1000 Sentences With "convex" | Random Sentence Generator

In particular, Pitt said, the GCI defines an interesting relationship between vectors on the surfaces of overlapping convex shapes, which could blossom into a new subdomain of convex geometry.

Relevance to Van Eyck becomes less convincing with the inclusion of the actual convex mirrors of William Orpen and Rossetti, the latter of whom owned twenty four mirrors, nine of them convex.

Kladrubers are distinguished by their convex head and prominent Roman nose.

I approach and am reflected, distorted in its crinkly, convex foil surfaces.

The Quest is notably heavier than the Rift, with a thick, convex front panel.

It supports a curved fiberglass plane whose convex underside is covered in plastic wrap.

In the elevator, I watch myself in the convex security mirror, my head ballooning.

Convex optimization techniques can then map the device representations to the incoming stream of data.

However, the experiment with the projector did show Bucky how convex lenses and projectors work.

Mr. Ashbery dedicated both "Flow Chart" and "Self-Portrait in a Convex Mirror" to Mr. Kermani.

Anxiety pricked at his pores, and he could hear his own heartbeat in his convex chest.

Users can tell them apart by feel: The convex one raises the table, the concave lowers it.

Caption: Designers hid two buttons (one convex, one concave) under each desk for raising and lowering it.

The small convex circus horses on a carved cake print (think shortbread) encapsulate his clean, rounding line.

Here's IMO 22 Problem 3: Let P = A1 A2 … Ak be a convex polygon on the plane.

On Duchamp's instructions, he photographed the concave plaster form upside-down after lighting it to appear convex.

No camera there, just a convex mirror as if to suggest that a camera isn't necessary anyway.

I mean, John Currin's "Nude in a Convex Mirror" is one of my all-time favorite paintings.

On Monday, the European South Observatory announced that it had successfully cast the largest convex mirror ever made.

We wanted to use Red Curve on the wall opposite Nude in Convex Mirror in Susan Morrow's Office.

The word convex has appeared in 10 New York Times articles in the past year, including on Jan.

The effect is kaleidoscopic as the constellation of convex pieces above reflects the movement of the visitors below.

The various curved forms of the "Rabbit" — head, torso and legs — function as a cascade of convex mirrors.

The result is an image that imperceptibly teeters between multiple angles, collapsing or bending over backwards into convex arcs.

Imagine two convex polygons, such as a rectangle and a circle, centered on a point that serves as the target.

And their tops are a convex mound, like a mushroom cap, so you can't pile anything on top of them.

However, given the unique convex track, the clever moon phase readout, and the low price this is worth a look.

When the pressure shifts, the bubbles became concave or convex, changing how light refracted through them and creating different colors.

However, work like Eversley's, known for his futuristic, machine-fabricated convex lenses, was not without criticism within the black community.

Why, when you saw a hollow mask from the inner, concave side, did it nonetheless look convex, like a face?

It's somewhat directional, but the hole in the middle and convex shape casts light at wider angles than other directional alarms.

If the post with the convex top suggests an umbrella that could protect someone against rain and sunlight, this does neither.

Sometimes using a camera, other times using a convex TV screen found in old junk shops to view the world through.

But for a longer read, we recommend "Self-Portrait in a Convex Mirror," from his 1975 collection of the same name.

There are all kind of weird theories I haven't looked into like hollow earth, convex earth—just stuff I don't look into.

My belly is convex, a sphere interrupted only by the rest of my body, and I put on whatever fits over it.

And it is indeed true that the shape of bi-convex lenses—the familiar sort used as magnifying glasses—resembles those leguminous seeds.

The more convex or "bowed out" the Lorenz curve, the more unequal the distribution of income, wealth or whatever else one is measuring.

Her exploration of space, the negative and positive attributes of convex and concave curves, is a major aesthetic topic in her Pelvis paintings.

One of the convex mesh plates that covered the left ear caved in when trapped for hours between my head and a pillow.

He had already made his mark as a youth with his "Self-Portrait in a Convex Mirror," which he presented to the pope.

In addition, the Tambour, like the new mechanical Tambour Moon, has a convex side case, which allows maximum face diameter with minimum weight.

In the case of John Currin's Nude in Convex Mirror, his gallery sent us a high-res digital file which we printed on canvas.

One particularly clever design survives from an eyeglasses store: the double-sided placard features golden lacquer frames embedded with convex, clear rock crystal lenses.

It's basically two convex walls with a gap between them, but the clip shows precisely how it could feel like a whole series of hallways.

The surface was gently bowed from top to bottom or from side to side, while all the edges were slightly curved — either concave or convex.

The concave and convex hexagon design isn't just there for aesthetics — the sculpted shapes are specifically placed to provide cushion and stability where it's needed most.

That means that while teaching new employees how to cook Arab flatbread on a saj (a convex griddle) and handling death threats, she had stomach-churning nausea.

And I signed up in the hopes it would be able to hold up a convex mirror to the web and finally bring my archive into view.

At least in my first two trimesters, when wearing anything I already owned was remotely possible, I wore shirts long ignored for their middle-clinginess, contentedly convex.

They often refined the experience using an invention known as the Claude glass, a small convex mirror mounted on black foil, which framed and "improved" nature's work.

Her drawings of birds, framed beneath convex glass, like Victorian specimens, are punctured with patterns of migration paths, food webs, and other charts sourced from scientific documents.

To enter the elevator, residents swing open the leaden door and as they board, perhaps they glance up at the convex mirror that reflects anyone behind them.

Ashbery-as-dorm-poster culminated with the cover shot by the artist Darragh Park for the Penguin paperback edition of "Self-Portrait in a Convex Mirror" (1975).

It also functioned as a self-portrait and hinted at Rejlander's hidden ambitions: reflected in the convex mirror, he presents himself as a modern-day Jan van Eyck.

The laser works on "slightly convex or concave" surfaces, Hung said, as long as the objects are placed 15-16 cm (5.9-6.2 inches) from the laser source.

"California Common Law" (2018) includes a convex mirror in the corner — a homage to Jan van Eyck's "Arnolfini Wedding" (1434), which famously played tricks with mirrors and windows.

After Jensdotter removed shavings of paint with a convex blade, the painting is largely black with faint flares of grey and orange emerging from layers below the surface.

And he had seen them with an instrument he had devised himself, trying out concave and convex lenses and, in the end, grinding his own on a rotary lathe.

We just started throwing all kinds of convex shapes onto the computer, and we just kept getting a crystal structure after another after another, some that were very complicated.

Everything else, from the joysticks, one convex and the other concave, to the silly placement of the extra black and white buttons, is the same as the original design.

We'll stick with "convex" polygons, those whose interior angles are each less than 180 degrees, and we'll allow ourselves to move them around, rotate them and flip them over.

"For this piece, the sapphire is even more convex than on previous versions in order to give even more volume to the globe," Mr. Ferrier said in an email.

A luxury item in the late Roman Empire, the cage cup has a convex bottom with no base, and nests inside an elaborate, completely detachable openwork stand (the "cage").

Its legs had been shortened, the shell was gouged and dirty, convex Perspex windows had been removed, and the interior had been flooded by rainwater and slathered in plaster.

Peering down the viewer's small convex lens, I saw Jacobs — gangly, laughing, 17 — standing next to his white-haired grandmother, herself the picture of bourgeois Upper West Side elegance.

Ashbery, who was first associated with the New York school of poetry in the 1950s and 1960s, won the Pulitzer Prize for "Self-Portrait in a Convex Mirror" in 1976.

Inside, Genesis has installed a 21-inch floating curved 4K screen from LG. The screen is convex in front of the driver, where it plays the role of instrument cluster.

The convex conjugate of a closed convex function is again a closed convex function. The convex conjugate of a polyhedral convex function (a convex function with polyhedral epigraph) is again a polyhedral convex function.

Eyes less convex. Pronotum moderately convex. Elytra oblong and moderately convex.

A 3-dimensional convex polytope. Convex analysis includes not only the study of convex subsets of Euclidean spaces but also the study of convex functions on abstract spaces. Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.

Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets. The branch of mathematics devoted to the study of properties of convex sets and convex functions is called convex analysis. The notion of a convex set can be generalized as described below.

The orthogonal convex hull is also known as the rectilinear convex hull, or, in two dimensions, the - convex hull.

In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex.

The convex hull of the red set is the blue and red convex set. In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset.

Extensions of convex optimization include the optimization of biconvex, pseudo-convex, and quasiconvex functions. Extensions of the theory of convex analysis and iterative methods for approximately solving non-convex minimization problems occur in the field of generalized convexity, also known as abstract convex analysis.

This preference relation is convex, but not strictly-convex. 3\. A preference relation represented by linear utility functions is convex, but not strictly convex. Whenever x\sim y, every convex combination of x,y is equivalent to any of them. 4\. Consider a preference relation represented by: :u(x_1,x_2) = \max(x_1,x_2) This preference relation is not convex.

A convex curve is the boundary of a convex set. A parabola, a simple example of a convex curve In geometry, a convex curve is a simple curve in the Euclidean plane which lies completely on one side of each and every one of its tangent lines. The boundary of a convex set is always a convex curve.

However, polynomially convex sets do not behave as nicely as convex sets. Kallin studied conditions under which unions of convex balls are polynomially convex, and found an example of three disjoint cubical cylinders whose union is not polynomially convex.. As part of her work on polynomial convexity, she proved a result now known as Kallin's lemma, giving conditions under which the union of two polynomially convex sets remains itself polynomially convex...

The boundary of a convex set is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set.

For example, the convex hull of any set of three distinct points forms a solid (i.e. "filled") triangle (including the perimeter). Also, in the plane , the unit circle is not convex but the closed unit disk is convex and furthermore, this disk is equal to the convex hull of the circle. :Definition: The closed convex hull of a set is the smallest closed and convex set containing .

Let be a set in a real or complex vector space. is star convex (star- shaped) if there exists an in such that the line segment from to any point in is contained in . Hence a non-empty convex set is always star-convex but a star-convex set is not always convex.

The convex hull or lower convex envelope of a function f on a real vector space is the function whose epigraph is the lower convex hull of the epigraph of f. It is the unique maximal convex function majorized by f. The definition can be extended to the convex hull of a set of functions (obtained from the convex hull of the union of their epigraphs, or equivalently from their pointwise minimum) and, in this form, is dual to the convex conjugate operation.

Let X,Y, and Z be topological vector spaces, S \subseteq X, T \subseteq Y, and A \subseteq X \times Y. The following implications hold: :complete \implies cs-complete \implies cs-closed \implies lower cs-closed (lcs-closed) and ideally convex. :lower cs-closed (lcs-closed) or ideally convex \implies lower ideally convex (li-convex) \implies convex. :(Hx) \implies (Hwx) \implies convex. The converse implication do not hold in general.

The collection of convex subsets of a vector space, an affine space, or a Euclidean space has the following properties:Soltan, Valeriu, Introduction to the Axiomatic Theory of Convexity, Ştiinţa, Chişinău, 1984 (in Russian). #The empty set and the whole space are convex. #The intersection of any collection of convex sets is convex. #The union of a sequence of convex sets is convex, if they form a non-decreasing chain for inclusion.

A set is convex if it contains all convex combinations of its points. The convex hull of a given set of points is identical to the set of all their convex combinations. There exist subsets of a vector space that are not closed under linear combinations but are closed under convex combinations. For example, the interval [0,1] is convex but generates the real-number line under linear combinations.

One of the widely used convex optimization algorithms is projections onto convex sets (POCS). This algorithm is employed to recover/synthesize a signal satisfying simultaneously several convex constraints. Let f_i be the indicator function of non-empty closed convex set C_i modeling a constraint. This reduces to convex feasibility problem, which require us to find a solution such that it lies in the intersection of all convex sets C_i.

In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming, probability theory, game theory, etc.

A strictly convex curve is a convex curve that does not contain any line segments. Equivalently, a strictly convex curve is a curve that intersects any line in at most two points,.. or a simple curve in convex position, meaning that none of its points is a convex combination of any other subset of its points.

If g_1 is K-convex and g_2 is L-convex, then for \alpha \geq 0, \beta \geq 0,\; g=\alpha g_1 +\beta g_2 is (\alpha K+\beta L)-convex.

Non‑convex phenomena in economics have been studied with nonsmooth analysis, which generalizes convex analysis.

Each convex set containing X must (by the assumption that it is convex) contain all convex combinations of points in X, so the set of all convex combinations is contained in the intersection of all convex sets containing X. Conversely, the set of all convex combinations is itself a convex set containing X, so it also contains the intersection of all convex sets containing X, and therefore the second and third definitions are equivalent., p. 12; , p. 17. In fact, according to Carathéodory's theorem, if X is a subset of a d-dimensional Euclidean space, every convex combination of finitely many points from X is also a convex combination of at most d+1 points in X. The set of convex combinations of a (d+1)-tuple of points is a simplex; in the plane it is a triangle and in three-dimensional space it is a tetrahedron.

The spire is depressed conoidal. Its outlines are convex, lower than the aperture. The yellowish- white protoconch consists of 2 convex smooth whorls. The 3½ convex whorls are rapidly increasing.

Given a convex shape (light blue) and its set of extreme points (red), the convex hull of is . In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about compact convex sets in locally convex topological vector spaces (TVSs). This theorem generalizes to infinite- dimensional spaces and to arbitrary compact convex sets the following basic observation: a convex (i.e. "filled") triangle, including its perimeter and the area "inside of it", is equal to the convex hull of its three vertices, where these vertices are exactly the extreme points of this shape.

In mathematical analysis (in particular convex analysis) and optimization, a proper convex function is a convex function f taking values in the extended real number line such that :f(x) < +\infty for at least one x and :f(x) > -\infty for every x. That is, a convex function is proper if its effective domain is nonempty and it never attains -\infty. Convex functions that are not proper are called improper convex functions. A proper concave function is any function g such that f = -g is a proper convex function.

Although , the convex balanced hull of is not necessarily equal to the balanced hull of the convex hull of . For an example where , let be the real vector space and let }. Then is a strict subset of cobal(S) that is not even convex. In particular, this example also shows that the balanced hull of a convex set is not necessarily convex.

In economics, convex hulls can be used to apply methods of convexity in economics to non-convex markets. In geometric modeling, the convex hull property Bézier curves helps find their crossings, and convex hulls are part of the measurement of boat hulls. And in the study of animal behavior, convex hulls are used in a standard definition of the home range.

For every proper convex function f on Rn there exist some b in Rn and β in R such that :f(x) \ge x \cdot b - \beta for every x. The sum of two proper convex functions is convex, but not necessarily proper. For instance if the sets A \subset X and B \subset X are non-empty convex sets in the vector space X, then the characteristic functions I_A and I_B are proper convex functions, but if A \cap B = \emptyset then I_A + I_B is identically equal to +\infty. The infimal convolution of two proper convex functions is convex but not necessarily proper convex..

For these graphs, a convex (but not necessarily strictly convex) drawing can be found within a grid whose length on each side is linear in the number of vertices of the graph, in linear time. However, strictly convex drawings may require larger grids; for instance, for any polyhedron such as a pyramid in which one face has a linear number of vertices, a strictly convex drawing of its graph requires a grid of cubic area. A linear-time algorithm can find strictly convex drawings of polyhedral graphs in a grid whose length on each side is quadratic. Convex but not strictly convex drawing of the complete bipartite graph K_{2,3} Other graphs that are not polyhedral can also have convex drawings, or strictly convex drawings.

Some graphs, such as the complete bipartite graph K_{2,3}, have convex drawings but not strictly convex drawings. A combinatorial characterization for the graphs with convex drawings is known, and they can be recognized in linear time, but the grid dimensions needed for their drawings and an efficient algorithm for constructing small convex grid drawings of these graphs are not known in all cases. Convex drawings should be distinguished from convex embeddings, in which each vertex is required to lie within the convex hull of its neighboring vertices. Convex embeddings can exist in dimensions other than two, do not require their graph to be planar, and even for planar embeddings of planar graphs do not necessarily force the outer face to be convex.

The acute spire is elevated conical, with sides slightly convex. The protoconch is conic with 1½ strongly convex smooth whorls, which are mostly pearly. The six whorls of the spire are flatly convex, the last angled at the periphery. The base of the shell is slightly convex.

A set that is not convex is called a non-convex set. A polygon that is not a convex polygon is sometimes called a concave polygon,. and some sources more generally use the term concave set to mean a non-convex set, but most authorities prohibit this usage. The complement of a convex set, such as the epigraph of a concave function, is sometimes called a reverse convex set, especially in the context of mathematical optimization..

A convex curve may be defined as the boundary of a convex set in the Euclidean plane. This definition is more restrictive than the definition in terms of tangent lines; in particular, with this definition, a convex curve can have no endpoints.. Sometimes, a looser definition is used, in which a convex curve is a curve that forms a subset of the boundary of a convex set. For this variation, a convex curve may have endpoints.

The classical orthogonal convex hull can be equivalently defined as the smallest orthogonally convex superset of a set K \subset \R^2, by analogy to the following definition of the convex hull: the convex hull of K is the smallest convex superset of K. The classical orthogonal convex hull might be disconnected. If a point set has no pair of points on a line parallel to one of the standard basis vectors, the classical orthogonal convex hull of such point set is equal to the point set itself. A well known property of convex hulls is derived from the Carathéodory's theorem: A point x \in \R^d is in the interior of the convex hull of a point set K \subset \R^d if, and only if, it is already in the convex hull of d+1 or fewer points of K. This property is also valid for classical orthogonal convex hulls.

An alternative method of making non- convex polygons convex that has also been studied is to perform flipturns, 180-degree rotations of a pocket around the midpoint of its convex hull edge.

The relative convex hull of a finite set of points in a simple polygon In discrete geometry and computational geometry, the relative convex hull or geodesic convex hull is an analogue of the convex hull for the points inside a simple polygon or a rectifiable simple closed curve.

A set is absolutely convex if it is convex and balanced. The convex subsets of (the set of real numbers) are the intervals and the points of . Some examples of convex subsets of the Euclidean plane are solid regular polygons, solid triangles, and intersections of solid triangles. Some examples of convex subsets of a Euclidean 3-dimensional space are the Archimedean solids and the Platonic solids.

In the convex hull of the red set, each blue point is a convex combination of some red points. In a real vector space, a set is defined to be convex if, for each pair of its points, every point on the line segment that joins them is covered by the set. For example, a solid cube is convex; however, anything that is hollow or dented, for example, a crescent shape, is non‑convex. Trivially, the empty set is convex.

A set of convex-shaped indifference curves displays convex preferences: Given a convex indifference curve containing the set of all bundles (of two or more goods) that are all viewed as equally desired, the set of all goods bundles that are viewed as being at least as desired as those on the indifference curve is a convex set. Convex preferences with their associated convex indifference mapping arise from quasi-concave utility functions, although these are not necessary for the analysis of preferences.

Exact formulas are known for enumerating polyominoes of special classes, such as the class of convex polyominoes and the class of directed polyominoes. The definition of a convex polyomino is different from the usual definition of convexity, but is similar to the definition used for the orthogonal convex hull. A polyomino is said to be vertically or column convex if its intersection with any vertical line is convex (in other words, each column has no holes). Similarly, a polyomino is said to be horizontally or row convex if its intersection with any horizontal line is convex.

If the objective function is concave (maximization problem), or convex (minimization problem) and the constraint set is convex, then the program is called convex and general methods from convex optimization can be used in most cases. If the objective function is quadratic and the constraints are linear, quadratic programming techniques are used. If the objective function is a ratio of a concave and a convex function (in the maximization case) and the constraints are convex, then the problem can be transformed to a convex optimization problem using fractional programming techniques. Several methods are available for solving nonconvex problems.

Also, products of convex spaces are still convex. This follows from the Kunneth theorem in coherent sheaf cohomology.

If instead of the sublinear property,R is convex, then R is a set-valued convex risk measure.

Any such polygonal chain has the same length, so there are infinitely many connected orthogonal convex hulls for the point set. For point sets in the plane, the connected orthogonal convex hull can be easily obtained from the maximal orthogonal convex hull. If the maximal orthogonal convex hull of a point set K \subset \R^2 is connected, then it is equal to the connected orthogonal convex hull of K. If this is not the case, then there are infinitely many connected orthogonal convex hulls for K, and each one can be obtained by joining the connected components of the maximal orthogonal convex hull of K with orthogonally convex alternating polygonal chains with interior angle 90^\circ.

In the field of convex optimization, there is an analogous statement which asserts that the maximum of a convex function on a compact convex set is attained on the boundary.Chapter 32 of Rockafellar (1970).

Fenchel lectured on "Convex Sets, Cones, and Functions" at Princeton University in the early 1950s. His lecture notes shaped the field of convex analysis, according to the monograph Convex Analysis of R. T. Rockafellar.

Conic optimization is a subfield of convex optimization that studies problems consisting of minimizing a convex function over the intersection of an affine subspace and a convex cone. The class of conic optimization problems includes some of the most well known classes of convex optimization problems, namely linear and semidefinite programming.

That is, is convex if and only if for all in , implies . A convex set is not connected in general: a counter-example is given by the space , which is both convex and totally disconnected.

The protoconch is small, globular and consists of one smooth and convex whorl. The four whorls increase rather rapidly. They are somewhat flattened below the suture, then convex. The base of the shell is convex.

The spire is conical, as high as the aperture and a little convex. The apex is acute. The protoconch is very small, consisting of 1½ smooth, slightly convex whorls. The six whorls are slightly convex.

The convex hull of a simple polygon is itself a convex polygon. Overlaying the original simple polygon onto its convex hull partitions this convex polygon into regions, one of which is the original polygon. The remaining regions are called pockets. Each pocket is itself a simple polygon, bounded by a polygonal chain on the boundary of the given simple polygon and by a single edge of the convex hull.

Reinhardt had previously considered the question of anisohedral convex polygons, showing that there were no anisohedral convex hexagons but being unable to show there were no such convex pentagons, while finding the five types of convex pentagon tiling the plane isohedrally. Kershner gave three types of anisohedral convex pentagon in 1968; one of these tiles using only direct isometries without reflections or glide reflections, so answering a question of Heesch.

Convex function on an interval. graph (in green) is a convex set. bivariate convex function . In mathematics, a real-valued function defined on an n-dimensional interval is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points.

The closed convex hull of a set is the closure of the convex hull, and the open convex hull is the interior (or in some sources the relative interior) of the convex hull. The closed convex hull of X is the intersection of all closed half-spaces containing X. If the convex hull of X is already a closed set itself (as happens, for instance, if X is a finite set or more generally a compact set), then it equals the closed convex hull. However, an intersection of closed half-spaces is itself closed, so when a convex hull is not closed it cannot be represented in this way. If the open convex hull of a set X is d-dimensional, then every point of the hull belongs to an open convex hull of at most 2d points of X. The sets of vertices of a square, regular octahedron, or higher-dimensional cross-polytope provide examples where exactly 2d points are needed.

For this property, the restriction to chains is important, as the union of two convex sets need not be convex.

The convex layers of a point set and their intersection with a halfplane In computational geometry, the convex layers of a set of points in the Euclidean plane are a sequence of nested convex polygons having the points as their vertices. The outermost one is the convex hull of the points and the rest are formed in the same way recursively. The innermost layer may be degenerate, consisting only of one or two points. The problem of constructing convex layers has also been called onion peeling or onion decomposition.. Although constructing the convex layers by repeatedly finding convex hulls would be slower, it is possible to partition any set of n points into its convex layers in time O(n\log n).

Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis and discrete mathematics. It has close connections to convex analysis, optimization and functional analysis and important applications in number theory. Convex geometry dates back to antiquity. Archimedes gave the first known precise definition of convexity.

For every subset Q of a real vector space, its is the minimal convex set that contains Q. Thus Conv(Q) is the intersection of all the convex sets that cover Q. The convex hull of a set can be equivalently defined to be the set of all convex combinations of points in Q.

Universum museum in Mexico City A three-dimensional solid is a convex set if it contains every line segment connecting two of its points. A convex polyhedron is a polyhedron that, as a solid, forms a convex set. A convex polyhedron can also be defined as a bounded intersection of finitely many half-spaces, or as the convex hull of finitely many points. Important classes of convex polyhedra include the highly symmetrical Platonic solids, the Archimedean solids and their duals the Catalan solids, and the regular-faced Johnson solids.

It is also equal to the closure of the convex hull of and to the intersection of all closed convex subsets that contain . It is straightforward to show that the convex hull of the extreme points forms a subset of , so the main burden of the proof is to show that there are enough extreme points so that their convex hull covers all of . As a corollary, it follows that every non-empty compact convex subset of a Hausdorff locally convex TVS has extreme points (i.e. the set of its extreme points is not empty).

In the Arrow–Debreu model of general economic equilibrium, agents are assumed to have convex budget sets and convex preferences. These assumptions of convexity in economics can be used to prove the existence of an equilibrium. When actual economic data is non-convex, it can be made convex by taking convex hulls. The Shapley–Folkman theorem can be used to show that, for large markets, this approximation is accurate, and leads to a "quasi- equilibrium" for the original non-convex market.. See in particular Section 16.9, Non Convexity and Approximate Equilibrium, pp. 209–210.

Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems of computational geometry.

An extreme point of a convex set is a point in the set that does not lie on a line segment between any other two points of the same set. For a convex hull, every extreme point must be part of the given set, because otherwise it cannot be formed as a convex combination of given points. According to the Krein–Milman theorem, every compact convex set in a Euclidean space (or more generally in a locally convex topological vector space) is the convex hull of its extreme points.; , p. 43.

The definition of a convex set and a convex hull extends naturally to geometries which are not Euclidean by defining a geodesically convex set to be one that contains the geodesics joining any two points in the set.

The protoconch is small, acute, and consists of two convex, light- brown, and finely spirally striate whorls. The six whorls are flatly convex. The body whorl is keeled at the periphery. The base of the shell is convex.

The specific name is derived from Latin convexus (meaning convex) and refers to the sacculus which is strongly convex dorso-basally.

The functional orthogonal convex hull is not defined using properties of sets, but properties of functions about sets. Namely, it restricts the notion of convex function as follows. A function f: \R^d \rightarrow \R is called orthogonally convex if its restriction to each line parallel to a non-zero of the standard basis vectors is a convex function.

Convexity is an important topic in economics. In the Arrow–Debreu model of general economic equilibrium, agents have convex budget sets and convex preferences: At equilibrium prices, the budget hyperplane supports the best attainable indifference curve. The profit function is the convex conjugate of the cost function. Convex analysis is the standard tool for analyzing textbook economics.

Bornological spaces are exactly those locally convex spaces for every bounded linear operator into another locally convex space is necessarily bounded. That is, a locally convex TVS is a bornological space if and only if for every locally convex TVS , a linear operator is continuous if and only if it is bounded. Every normed space is bornological.

224 is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons. Forty-seven non-prismatic convex uniform 4-polytopes, one finite set of convex prismatic forms, and two infinite sets of convex prismatic forms have been described. There are also an unknown number of non-convex star forms.

Convex and strictly convex grid drawings of the same graph In graph drawing, a convex drawing of a planar graph is a drawing that represents the vertices of the graph as points in the Euclidean plane and the edges as straight line segments, in such a way that all of the faces of the drawing (including the outer face) have a convex boundary. The boundary of a face may pass straight through one of the vertices of the graph without turning; a strictly convex drawing asks in addition that the face boundary turns at each vertex. That is, in a strictly convex drawing, each vertex of the graph is also a vertex of each convex polygon describing the shape of each incident face. Every polyhedral graph has a strictly convex drawing, for instance obtained as the Schlegel diagram of a convex polyhedron representing the graph.

Various convex hull algorithms deal both with the facet enumeration and face lattice construction. In the planar case, i.e., for a convex polygon, both facet and vertex enumeration problems amount to the ordering vertices (resp. edges) around the convex hull.

The definition of relative convex hulls based on minimum enclosure does not extend to higher dimensions, because (even without being surrounded by an outer shape) the minimum surface area enclosure of a non-convex set is not generally convex. However, for the relative convex hull of a connected set within another set, a similar definition to one for simple polygons can be used. In this case, a relatively convex set can again be defined as a subset of the given outer set that contains all line segments in the outer set between pairs of its points. The relative convex hull can be defined as the intersection of all relatively convex sets that contain the inner set.

In contrast with the classical convexity where there exist several equivalent definitions of the convex hull, definitions of the orthogonal convex hull made by analogy to those of the convex hull result in different geometric objects. So far, researchers have explored the following four definitions of the orthogonal convex hull of a set K \subset \R^d: #Maximal definition: The definition described in the introduction of this article. It is based on the Maxima of a point set. #Classical definition: The orthogonal convex hull of K is the intersection of all orthogonally convex supersets of K; .

The orthogonal convex hull of a point set In geometry, a set is defined to be orthogonally convex if, for every line that is parallel to one of standard basis vectors, the intersection of with is empty, a point, or a single segment. The term "orthogonal" refers to corresponding Cartesian basis and coordinates in Euclidean space, where different basis vectors are perpendicular, as well as corresponding lines. Unlike ordinary convex sets, an orthogonally convex set is not necessarily connected. The orthogonal convex hull of a set is the intersection of all connected orthogonally convex supersets of .

The convex hull is commonly known as the minimum convex polygon in ethology, the study of animal behavior, where it is a classic, though perhaps simplistic, approach in estimating an animal's home range based on points where the animal has been observed., p. 137–140; Outliers can make the minimum convex polygon excessively large, which has motivated relaxed approaches that contain only a subset of the observations, for instance by choosing one of the convex layers that is close to a target percentage of the samples, or in the local convex hull method by combining convex hulls of neighborhoods of points.

The isoperimetric problem, a recurring concept in convex geometry, was studied by the Greeks as well, including Zenodorus. Archimedes, Plato, Euclid, and later Kepler and Coxeter all studied convex polytopes and their properties. From the 19th century on, mathematicians have studied other areas of convex mathematics, including higher-dimensional polytopes, volume and surface area of convex bodies, Gaussian curvature, algorithms, tilings and lattices.

A locally convex Hausdorff quasibarrelled space that is sequentially complete is barrelled. A locally convex Hausdorff quasibarrelled space is a Mackey space, quasi-M-barrelled, and countably quasibarrelled. A locally convex space is reflexive if and only if it is semireflexive and quasibarrelled. A locally convex quasi-barreled space that is also a 𝜎-barrelled space is a barrelled space.

Every convex curve that is the boundary of a closed convex set has a well-defined finite length. That is, these curves are a subset of the rectifiable curves. According to the four-vertex theorem, every smooth convex curve that is the boundary of a closed convex set has at least four vertices, points that are local minima or local maxima of curvature..

LSE is convex but not strictly convex. We can define a strictly convex log-sum-exp type function by adding an extra argument set to zero: :LSE_0^+(x_1,...,x_n) = LSE(0,x_1,...,x_n) This function is a proper Bregman generator (strictly convex and differentiable). It is encountered in machine learning, for example, as the cumulant of the multinomial/binomial family.

For relative convex hulls of simple polygons, an alternative but equivalent definition of convexity can be used. A simple polygon P within another simple polygon Q is relatively convex or Q-convex if every line segment contained in Q that connects two points of P lies within P. The relative convex hull of a simple polygon P within Q can be defined as the intersection of all Q-convex polygons that contain P, as the smallest Q-convex polygon that contains P, or as the minimum-perimeter simple polygon that contains P and is contained by Q. generalizes linear time algorithms for the convex hull of a simple polygon to the relative convex hull of one simple polygon within another. The resulting generalized algorithm is not linear time, however: its time complexity depends on the depth of nesting of certain features of one polygon within another. In this case, the relative convex hull is itself a simple polygon.

Let be a TVS (not necessarily Hausdorff or locally convex). :Definition: For any , the convex (resp. balanced, disked, closed convex, closed balanced, closed disked) hull of is the smallest subset of that has this property and contains . We denote the closure (resp.

It can be computed in linear time, faster than algorithms for convex hulls of point sets. The convex hull of a simple polygon can be subdivided into the given polygon itself and into polygonal pockets bounded by a polygonal chain of the polygon together with a single convex hull edge. Repeatedly reflecting an arbitrarily chosen pocket across this convex hull edge produces a sequence of larger simple polygons; according to the Erdős–Nagy theorem, this process eventually terminates with a convex polygon.

The convex hull of a simple polygon (blue). Its four pockets are shown in yellow; the whole region shaded in either color is the convex hull. In discrete geometry and computational geometry, the convex hull of a simple polygon is the polygon of minimum perimeter that contains a given simple polygon. It is a special case of the more general concept of a convex hull.

3D convex hull of 120 point cloud It is not obvious that the first definition makes sense: why should there exist a unique minimal convex set containing X, for every X? However, the second definition, the intersection of all convex sets containing X, is well-defined. It is a subset of every other convex set Y that contains X, because Y is included among the sets being intersected. Thus, it is exactly the unique minimal convex set containing X. Therefore, the first two definitions are equivalent.

These definitions are made by analogy with the classical theory of convexity, in which is convex if, for every line , the intersection of with is empty, a point, or a single segment. Orthogonal convexity restricts the lines for which this property is required to hold, so every convex set is orthogonally convex but not vice versa. For the same reason, the orthogonal convex hull itself is a subset of the convex hull of the same point set. A point belongs to the orthogonal convex hull of if and only if each of the closed axis-aligned orthants having as apex has a nonempty intersection with .

In economics, non-convexity refers to violations of the convexity assumptions of elementary economics. Basic economics textbooks concentrate on consumers with convex preferences (that do not prefer extremes to in-between values) and convex budget sets and on producers with convex production sets; for convex models, the predicted economic behavior is well understood. When convexity assumptions are violated, then many of the good properties of competitive markets need not hold: Thus, non-convexity is associated with market failures, where supply and demand differ or where market equilibria can be inefficient. Non-convex economies are studied with nonsmooth analysis, which is a generalization of convex analysis.

Since the (red) part of the (black and red) line-segment joining the points x and y lies outside of the (green) set, the set is non-convex. In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points, it contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersect every line into a single line segment (possibly empty). For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex.

An oloid, the convex hull of two circles in 3d space For the convex hull of a space curve or finite set of space curves in general position in three- dimensional space, the parts of the boundary away from the curves are developable and ruled surfaces. Examples include the oloid, the convex hull of two circles in perpendicular planes, each passing through the other's center, the sphericon, the convex hull of two semicircles in perpendicular planes with a common center, and D-forms, the convex shapes obtained from Alexandrov's uniqueness theorem for a surface formed by gluing together two planar convex sets of equal perimeter.

Its sides are slightly convex. The apex is subacute. The protoconch is conoidal, consisting of 3 convex spirally striate whorls. The whorls number 6 to 7.

There are six whorls, convex, the last one rounded and convex beneath. The suture is impressed. The aperture is oblique. The outer wall is moderately thick.

It follows from the above property that a convex cone can also be defined as a linear cone that is closed under convex combinations, or just under additions. More succinctly, a set C is a convex cone if and only if and , for any positive scalar α.

The convex whorls are encircled by stride. The body whorl is large, scarcely angulated. The base of the shell is a little convex. The suture is distinct.

Universum museum in Mexico City. The image splits between the convex and concave curves. A large convex mirror. Distortions in the image increase with the viewing distance.

Algorithms that construct convex hulls of various objects have a broad range of applications in mathematics and computer science. In computational geometry, numerous algorithms are proposed for computing the convex hull of a finite set of points, with various computational complexities. Computing the convex hull means that a non-ambiguous and efficient representation of the required convex shape is constructed. The complexity of the corresponding algorithms is usually estimated in terms of n, the number of input points, and sometimes also in terms of h, the number of points on the convex hull.

Convex mirror lets motorists see around a corner. Detail of the convex mirror in the Arnolfini Portrait The passenger-side mirror on a car is typically a convex mirror. In some countries, these are labeled with the safety warning "Objects in mirror are closer than they appear", to warn the driver of the convex mirror's distorting effects on distance perception. Convex mirrors are preferred in vehicles because they give an upright (not inverted), though diminished (smaller), image and because they provide a wider field of view as they are curved outwards.

In introductory level math books, the term convex is often conflated with the opposite term concave by referring to a "concave function" as "convex downward". Likewise, a "concave" function is referred to as "convex upwards" to distinguish it from "convex downwards". However, the use of "up" and "down" keyword modifiers is not universally used in the field of mathematics, and mostly exists to avoid confusing students with an extra term for concavity. If the term "convex" is used without an "up" or "down" keyword, then it refers strictly to a cup shaped graph \cup.

They can be solved in time O(n\log n) for two or three dimensional point sets, and in time matching the worst-case output complexity given by the upper bound theorem in higher dimensions. As well as for finite point sets, convex hulls have also been studied for simple polygons, Brownian motion, space curves, and epigraphs of functions. Convex hulls have wide applications in mathematics, statistics, combinatorial optimization, economics, geometric modeling, and ethology. Related structures include the orthogonal convex hull, convex layers, Delaunay triangulation and Voronoi diagram, and convex skull.

The figure shows a set of 16 points in the plane and the orthogonal convex hull of these points. As can be seen in the figure, the orthogonal convex hull is a polygon with some degenerate edges connecting extreme vertices in each coordinate direction. For a discrete point set such as this one, all orthogonal convex hull edges are horizontal or vertical. In this example, the orthogonal convex hull is connected.

The shell of C. depressa is generally flat and white, ranging from extremely recurved to somewhat convex depending on the habitat of the individual. Those from exposed substrates are often oval and convex. The septum is flat in convex shells and convex in recurved shells, with a notch on the right side where it attaches to the shell. There is also a depression in the center of the septal margin.

König eyepiece diagram The König eyepiece has a concave-convex positive doublet and a plano-convex singlet. The strongly convex surfaces of the doublet and singlet face and (nearly) touch each other. The doublet has its concave surface facing the light source and the singlet has its almost flat (slightly convex) surface facing the eye. It was designed in 1915 by German optician Albert König (1871−1946) as a simplified Abbe.

In some cases, a single flip will cause a non-convex simple polygon to become convex. Once this happens, no more flips are possible. The Erdős–Nagy theorem states that it is always possible to find a sequence of flips that produces a convex polygon in this way. More strongly, for every simple polygon, every sequence of flips will eventually produce a convex polygon, in a finite number of steps.

There are infinitely many deltahedra, but of these only eight are convex, having 4, 6, 8, 10, 12, 14, 16 and 20 faces. (They showed that there are just 8 convex deltahedra. ) The number of faces, edges, and vertices is listed below for each of the eight convex deltahedra.

Complete and metrizable TVSs are ultrabarrelled. If X is a complete locally bounded non-locally convex TVS and if B is a closed balanced and bounded neighborhood of the origin, then B is an ultrabarrel that is not convex and has a defining sequence consisting of non-convex sets.

Hence, this transformation transforms every GP into an equivalent convex program. In fact, this log-log transformation can be used to convert a larger class of problems, known as log-log convex programming (LLCP), into an equivalent convex form.A. Agrawal, S. Diamond, and S. Boyd. Disciplined Geometric Programming.

The definition using intersections of convex sets may be extended to non-Euclidean geometry, and the definition using convex combinations may be extended from Euclidean spaces to arbitrary real vector spaces or affine spaces; convex hulls may also be generalized in a more abstract way, to oriented matroids.

An affine convex cone is the set resulting from applying an affine transformation to a convex cone. A common example is translating a convex cone by a point p: p+C. Technically, such transformations can produce non-cones. For example, unless p=0, p+C is not a linear cone.

The length of the shell attains 5.25 mm, its diameter 2.25 mm. (Original description) The small, white, solid shell is shining, elongate-oval and blunt. It consists of 5 whorls, including a protoconch of 2 smooth convex whorls, and a very flatly convex apex. The spire-whorls are sloping convex.

A kinetic convex hull data structure is a kinetic data structure that maintains the convex hull of a set of continuously moving points. It should be distinguished from dynamic convex hull data structures, which handle points undergoing discrete changes such as insertions or deletions of points rather than continuous motion.

The protoconch is subacute. The spire has very convex outlines. It contains about six whorls, moderately convex or subangular, with a lightly impressed suture. The aperture is quite narrow.

The spire is elevated, sometimes scalariform. The apex acute. The upper whorls are slightly convex. The body whorl is convex, depressed below the suture and, rounded at the periphery.

The types treated here all give convex bounding volumes. If the object being bounded is known to be convex, this is not a restriction. If non-convex bounding volumes are required, an approach is to represent them as a union of a number of convex bounding volumes. Unfortunately, intersection tests become quickly more expensive as the bounding boxes become more sophisticated. A ' is a cuboid, or in 2-D a rectangle, containing the object.

In cluster analysis, squared distances can be used to strengthen the effect of longer distances. Squared Euclidean distance is not a metric, as it does not satisfy the triangle inequality. However it is a smooth, strictly convex function of the two points, unlike the distance, which is non-smooth for equal points and convex but not strictly convex. The squared distance is thus preferred in optimization theory, since it allows convex analysis to be used.

Postcoxal pits of metaventrite absent. Metaventrite flat to slightly convex, or moderately to strongly convex. Transverse groove of metaventrite absent. Anterior edge of metaventrite without transverse carina between mesocoxal cavities.

The spire is conical, its outlines a trifle concave;. It contains 8 whorls. The dextral apex is subimmersed. The first two whorls are quite convex, the following whorls slightly convex.

In a convex polygon, all interior angles are less than or equal to 180 degrees, while in a strictly convex polygon all interior angles are strictly less than 180 degrees.

In major workers, the clypeus is depressed and finely punctate. The anterior margin is convex. In minor workers, the clypeus is wide, and the anterior margin is convex and projecting.

The five convex whorls are subacute at the periphery. The suture is deep. The base of the shell is convex and the pale umbilicus is deep. The aperture is large.

The first one is rounded, the second whorl is slightly convex, the others are convex and obtusely angulated. They show a few smooth oblique plicae. The interstices between the ribs are rather smooth. The body whorl is oblong and slightly convex on top and with a short, but conspicuous, fold below the suture.

This geometrizes the axioms in terms of "sums with (possible) restrictions on the coordinates". Another concept from convex analysis is a convex function from to real numbers, which is defined through an inequality between its value on a convex combination of points and sum of values in those points with the same coefficients.

Boosting algorithms can be based on convex or non-convex optimization algorithms. Convex algorithms, such as AdaBoost and LogitBoost, can be "defeated" by random noise such that they can't learn basic and learnable combinations of weak hypotheses.P. Long and R. Servedio. 25th International Conference on Machine Learning (ICML), 2008, pp. 608--615.

See "Circle and B-Splines clipping algorithms" under the subject Clipping (computer graphics) for an example of use. A convex hull is the smallest convex volume containing the object. If the object is the union of a finite set of points, its convex hull is a polytope. A ' ('DOP) generalizes the bounding box.

Ben-Hain and ElishakoffBen Haim Y. and Elishakoff I., Convex Models of Uncertainty in Applied Mechanics, Elsevier Science Publishers, Amsterdam, 1990 (1990), Elishakoff et al.I. Elishakoff, I. Lin Y.K. and Zhu L.P., Probabilistic and Convex Modeling of Acoustically Excited Structures, Elsevier Science Publishers, Amsterdam, 1994 (1994) applied convex analysis to model uncertainty.

Adipose spine straight or slightly convex dorsally, membrane posteriorly convex. Pectoral-spine short, tip usually reaching the first quarter of pelvic spine, exceptionally extending up to the first third in large specimens (presumably males). Anal fin short with weak spine, its margin convex. Caudal fin slightly concave, ventral lobe longer than dorsal lobe.

Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. Well-known examples of convex functions of a single variable include the squaring function x^2 and the exponential function e^x. In simple terms, a convex function refers to a function that is in the shape of a cup \cup, and a concave function is in the shape of a cap \cap.

See the page on direction- preserving function for definitions. Continuous fixed-point theorems often require a convex set. The analogue of this property for discrete sets is an integrally-convex set.

The 5½ whorls are convex and tubular. The body whorl is slightly convex beneath and carinated around the umbilicus. The aperture is oblique, and circular. Its margins are thin and arcuate.

The five whorls are convex, slightly excavated at the sutures. They are nearly smooth and obsoletely spirally lirate. The large body whorl is convex below. The ovate aperture is silvery within.

The theorem holds only for sets that are compact (thus, in particular, bounded and closed) and convex (or homeomorphic to convex). The following examples show why the pre-conditions are important.

The basal shafts of the veins are convex, but each vein forks distally into an anterior convex branch and a posterior concave branch. Thus the costa and subcosta are regarded as convex and concave branches of a primary first vein, Rs is the concave branch of the radius, posterior media the concave branch of the media, Cu1 and Cu2 are respectively convex and concave, while the primitive Postcubitus and the first vannal have each an anterior convex branch and a posterior concave branch. The convex or concave nature of the veins has been used as evidence in determining the identities of the persisting distal branches of the veins of modern insects, but it has not been demonstrated to be consistent for all wings.

If is a TVS (over ℝ or ℂ) then a half-space is a set of the form for some real and some continuous real linear functional on . Note that the above theorem implies that the closed and convex subsets of a locally convex space depend entirely on the continuous dual space. Consequently, the closed and convex subsets are the same in any topology compatible with duality; that is, if and are any locally convex topologies on with the same continuous dual spaces, then a convex subset of is closed in the topology if and only if it is closed in the topology. This implies that the -closure of any convex subset of is equal to its -closure and that for any -closed disk in , .

In a linear programming problem, a series of linear constraints produce a convex feasible region of possible values for those variables. In the two-variable case this region is in the shape of a convex simple polygon. A convex feasible set is one in which a line segment connecting any two feasible points goes through only other feasible points, and not through any points outside the feasible set. Convex feasible sets arise in many types of problems, including linear programming problems, and they are of particular interest because, if the problem has a convex objective function that is to be maximized, it will generally be easier to solve in the presence of a convex feasible set and any local optimum will also be a global optimum.

Günter M. Ziegler introduces oriented matroids via convex polytopes.

The operations of constructing the convex hull and taking the Minkowski sum commute with each other, in the sense that the Minkowski sum of convex hulls of sets gives the same result as the convex hull of the Minkowski sum of the same sets. This provides a step towards the Shapley–Folkman theorem bounding the distance of a Minkowski sum from its convex hull., Theorem 3, pages 562–563; , Theorem 1.1.2 (pages 2–3) and Chapter 3.

From top to bottom, the second to the fourth figures show respectively, the maximal, the connected, and the functional orthogonal convex hull of the point set. As can be seen, the orthogonal convex hull is a polygon with some degenerate "edges", namely, orthogonally convex alternating polygonal chains with interior angle 90^\circ connecting extreme vertices.

Tangent bundle Totally convex. A subset K of a Riemannian manifold M is called totally convex if for any two points in K any geodesic connecting them lies entirely in K, see also convex. Totally geodesic submanifold is a submanifold such that all geodesics in the submanifold are also geodesics of the surrounding manifold.

Illustration of a convex set which looks somewhat like a deformed circle. The (black) line segment joining points x and y lies completely within the (green) set. Since this is true for any points x and y within the set that we might choose, the set is convex. Illustration of a non-convex set.

For all Hermitian × matrices and and all differentiable convex functions : ℝ → ℝ with derivative , or for all positive-definite Hermitian × matrices and , and all differentiable convex functions :(0,∞) → ℝ, the following inequality holds, In either case, if is strictly convex, equality holds if and only if = . A popular choice in applications is , see below.

However, this may not be true for convex sets that are not compact; for instance, the whole Euclidean plane and the open unit ball are both convex, but neither one has any extreme points. Choquet theory extends this theory from finite convex combinations of extreme points to infinite combinations (integrals) in more general spaces.

Let be a convex balanced neighborhood of 0 in a locally convex topological vector space and suppose is not an element of . Then there exists a continuous linear functional on such that : .

By definition, the connected orthogonal convex hull is always connected. However, it is not unique. Consider for example a pair of points in the plane not lying on an horizontal or a vertical line. The connected orthogonal convex hull of such points is an orthogonally convex alternating polygonal chain with interior angle 90^\circ connecting the points.

A Connected Ortho-convex Hull of the point set of the top figure. It is formed by the point set, the colored area and the two ortho- convex polygonal chains. The Functional Ortho-convex Hull of the point set of the top figure. It is formed by the point set, the colored area, and the four line segments.

The protoconch consists of two flatly convex whorls, which are finely spirally lirate with very distinct oblique growth-lines. The 4 to 5 whorls are slightly convex. The body whorl is large, concave below the suture, obtusely angulate at the periphery and eroded in front of the aperture. The base of the shell is flatly convex.

A polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram. The outer vertices of a compound can be connected to form a convex polyhedron called its convex hull. A compound is a facetting of its convex hull.

The condition of locally convex was added later by Nicolas Bourbaki. It's important to note that a sizable number of authors (e.g. Schaefer) use "F-space" to mean a (locally convex) Fréchet space while others do not require that a "Fréchet space" be locally convex. Moreover, some authors even use "F-space" and "Fréchet space" interchangeably.

A polygonal pseudotriangle is a polygon that has exactly three convex vertices. In particular, any triangle, and any nonconvex quadrilateral, is a pseudotriangle. The convex hull of any pseudotriangle is a triangle. The curves along the pseudotriangle boundary between each pair of convex vertices either lie within the triangle or coincide with one of its edges.

In convex analysis and the calculus of variations, branches of mathematics, a pseudoconvex function is a function that behaves like a convex function with respect to finding its local minima, but need not actually be convex. Informally, a differentiable function is pseudoconvex if it is increasing in any direction where it has a positive directional derivative.

Suppose is a vector space over , a subfield of the complex numbers (normally itself or ). A locally convex space is defined either in terms of convex sets, or equivalently in terms of seminorms.

The orange disc (a deeper colour than the thallus) surface is concave initially but matures to convex; the apothecium rim also narrows giving the effect that the convex disc is spilling over it.

But y is not in ch(X ∩ near(y)). See image at the right. Therefore X is not integrally-convex. In contrast, the set Y = { (0,0), (1,0), (2,0), (1,1), (2,1) } is integrally-convex.

A similar method can be used to subdivide squares into four congruent non-convex pentagons, or regular hexagons into six congruent non-convex pentagons, and then tile the plane with the resulting unit.

There are many examples of convex spaces, including the following.

In Erythrosuchus, the margin is convex and lacks a hook.

The convex to flattened cap is up to in diameter.

In the mathematical field of graph theory, a convex bipartite graph is a bipartite graph with specific properties. A bipartite graph, (U ∪ V, E), is said to be convex over the vertex set U if U can be enumerated such that for all v ∈ V the vertices adjacent to v are consecutive. Convexity over V is defined analogously. A bipartite graph (U ∪ V, E) that is convex over both U and V is said to be biconvex or doubly convex.

Blaschke sums can be used to decompose polytopes into simpler polytopes. In particular, every d-dimensional convex polytope with n facets can be represented as a Blaschke sum of at most n-d simplices (not necessarily of the same dimension). Every d-dimensional centrally symmetric convex polytope can be represented as a Blaschke sum of parallelotopes. And every d-dimensional convex polytope can be represented as a Blaschke sum of d-dimensional convex polytopes, each having at most 2d facets.

Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets. Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard. Sahni, S. "Computationally related problems," in SIAM Journal on Computing, 3, 262--279, 1974.Quadratic programming with one negative eigenvalue is NP-hard, Panos M. Pardalos and Stephen A. Vavasis in Journal of Global Optimization, Volume 1, Number 1, 1991, pg.15-22.

Convex hulls have wide applications in many fields. Within mathematics, convex hulls are used to study polynomials, matrix eigenvalues, and unitary elements, and several theorems in discrete geometry involve convex hulls. They are used in robust statistics as the outermost contour of Tukey depth, are part of the bagplot visualization of two-dimensional data, and define risk sets of randomized decision rules. Convex hulls of indicator vectors of solutions to combinatorial problems are central to combinatorial optimization and polyhedral combinatorics.

A bagplot. The outer shaded region is the convex hull, and the inner shaded region is the 50% Tukey depth contour. In robust statistics, the convex hull provides one of the key components of a bagplot, a method for visualizing the spread of two- dimensional sample points. The contours of Tukey depth form a nested family of convex sets, with the convex hull outermost, and the bagplot also displays another polygon from this nested family, the contour of 50% depth.

According to the above definition, if C is a convex cone, then C ∪ {0} is a convex cone, too. A convex cone is said to be ' if 0 is in C, and ' if 0 is not in C. Blunt cones can be excluded from the definition of convex cone by substituting "non-negative" for "positive" in the condition of α, β. A cone is called flat if it contains some nonzero vector x and its opposite -x, meaning C contains a linear subspace of dimension at least one, and salient otherwise. A blunt convex cone is necessarily salient, but the converse is not necessarily true.

The selected representation of the convex hull may influence on the computational complexity of further operations of the overall algorithm. For example, the point in polygon query for a convex polygon represented by the ordered set of its vertices may be answered in logarithmic time, which would be impossible for convex hulls reported by the set of it vertices without any additional information. Therefore, some research of dynamic convex hull algorithms involves the computational complexity of various geometric search problems with convex hulls stored in specific kinds of data structures. The mentioned approach of Overmars and van Leeuwen allows for logarithmic complexity of various common queries.

Bregman divergences correspond to convex functions on convex sets. Given a strictly convex, continuously-differentiable function on a convex set, known as the Bregman generator, the Bregman divergence measures the convexity of: the error of the linear approximation of from as an approximation of the value at : :D_F(p, q) = F(p)-F(q)-\langle abla F(q), p-q\rangle. The dual divergence to a Bregman divergence is the divergence generated by the convex conjugate of the Bregman generator of the original divergence. For example, for the squared Euclidean distance, the generator is , while for the relative entropy the generator is the negative entropy .

Convex hull of a simple polygon The convex hull of a simple polygon encloses the given polygon and is partitioned by it into regions, one of which is the polygon itself. The other regions, bounded by a polygonal chain of the polygon and a single convex hull edge, are called pockets. Computing the same decomposition recursively for each pocket forms a hierarchical description of a given polygon called its convex differences tree. Reflecting a pocket across its convex hull edge expands the given simple polygon into a polygon with the same perimeter and larger area, and the Erdős–Nagy theorem states that this expansion process eventually terminates.

Therefore, in the general case the convex hull of n points cannot be computed more quickly than sorting. The standard Ω(n log n) lower bound for sorting is proven in the decision tree model of computing, in which only numerical comparisons but not arithmetic operations can be performed; however, in this model, convex hulls cannot be computed at all. Sorting also requires Ω(n log n) time in the algebraic decision tree model of computation, a model that is more suitable for convex hulls, and in this model convex hulls also require Ω(n log n) time.Preparata, Shamos, Computational Geometry, Chapter "Convex Hulls: Basic Algorithms" However, in models of computer arithmetic that allow numbers to be sorted more quickly than O(n log n) time, for instance by using integer sorting algorithms, planar convex hulls can also be computed more quickly: the Graham scan algorithm for convex hulls consists of a single sorting step followed by a linear amount of additional work.

An acceptance set is convex (coherent) if and only if the corresponding risk measure is convex (coherent). As defined below it can be shown that R_{A_R}(X) = R(X) and A_{R_A} = A.

The topology on can be described by specifying that an absolutely convex subset is a neighborhood of 0 if and only if U \cap X_n is an absolutely convex neighborhood of in for every n.

In particular, in median graphs, the convex subgraphs have the Helly property: if a family of convex subgraphs has the property that all pairwise intersections are nonempty, then the whole family has a nonempty intersection.

The body is oval, although the ventral is not prominently convex.

The forewings are longer and the outer margin is more convex.

A complete ulna shows a slender bone and convex distal ends.

However, this is false; there are smaller non-convex Kakeya sets.

One method is to define a minimum district to convex polygon ratio . To use this method, every proposed district is circumscribed by the smallest possible convex polygon (its convex hull; think of stretching a rubberband around the outline of the district). Then, the area of the district is divided by the area of the polygon; or, if at the edge of the state, by the portion of the area of the polygon within state boundaries. Minimal convex polygon, showing how to rate district shape irregularity.

A convex polytope may be defined in a number of ways, depending on what is more suitable for the problem at hand. Grünbaum's definition is in terms of a convex set of points in space. Other important definitions are: as the intersection of half-spaces (half-space representation) and as the convex hull of a set of points (vertex representation).

A set C in a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk. The disked hull or the absolute convex hull of a set is the intersection of all disks containing that set.

A demo of Graham's scan to find a 2D convex hull. Graham's scan is a method of finding the convex hull of a finite set of points in the plane with time complexity O(n log n). It is named after Ronald Graham, who published the original algorithm in 1972. The algorithm finds all vertices of the convex hull ordered along its boundary.

BrownBoost uses a non-convex potential loss function, thus it does not fit into the AdaBoost framework. The non-convex optimization provides a method to avoid overfitting noisy data sets. However, in contrast to boosting algorithms that analytically minimize a convex loss function (e.g. AdaBoost and LogitBoost), BrownBoost solves a system of two equations and two unknowns using standard numerical methods.

In mathematics, Loewner order is the partial order defined by the convex cone of positive semi-definite matrices. This order is usually employed to generalize the definitions of monotone and concave/convex scalar functions to monotone and concave/convex Hermitian valued functions. These functions arise naturally in matrix and operator theory and have applications in many areas of physics and engineering.

The shell is very small, and it is one of the smallest of the endemic species found in the Cape Verde Islands. General profile is ventricosely conical, somewhat elongated with a rounded shoulder. Spire moderate, straight to slightly convex with 4-5 well defined cords on the flat to slightly convex sutural ramps. Sides of the last whorl are straight or slightly convex.

In standard NMF, matrix factor ， i.e., can be anything in that space. Convex NMFC Ding, T Li, MI Jordan, Convex and semi-nonnegative matrix factorizations, IEEE Transactions on Pattern Analysis and Machine Intelligence, 32, 45-55, 2010 restricts the columns of to convex combinations of the input data vectors (v_1, \cdots, v_n) . This greatly improves the quality of data representation of .

The shell contains 6½ whorls. The 2½ protoconch whorls are convex and smooth. The third whorl is convex and is slightly obliquely lirate. The others have a broad sloping shoulder and a greatly rounded anterior portion.

Its upper margin is convex, the basal one nearly straight. The columellar marginis slightly convex, reflected above. The parietal wall has a thin layer of enamel.Schepman 1908–1913, The Prosobranchia of the Siboga Expedition; Leyden,E.

94-104Dieudonne, J. History of Functional Analysis Chapter VIII. Section 1. Finally, in 1935 von Neumann introduced the general definition of a locally convex space (called a convex space by him).von Neumann, J. Collected works.

With Barrett O'Neill he made foundational contributions to the study of convex functions and convex sets in Riemannian geometry and their applications in the study of negative sectional curvature, including to the geometry of warped products.

They are slightly convex, the last rounded or obtusely angular. The base of the shell is flatly convex. The suture is linear, slightly impressed. The aperture is less than one-half the length of the shell.

The spire is conical, and rather obtuse. The protoconch is minute, strongly convex, smooth. The shell has 4-5 convex whorls. The last whorl is slightly greater than one-third of the height of the shell.

An antipodal pair of a convex polygon is a pair of 2 points admitting 2 infinite parallel lines being tangent to both points included in the antipodal without crossing any other line of the convex polygon.

One can show that the Brunn–Minkowski inequality for convex bodies in Rn implies Minkowski's first inequality for convex bodies in Rn, and that equality in the Brunn–Minkowski inequality implies equality in Minkowski's first inequality.

Economists have increasingly studied non-convex sets with nonsmooth analysis, which generalizes convex analysis. Convex analysis centers on convex sets and convex functions, for which it provides powerful ideas and clear results, but it is not adequate for the analysis of non-convexities, such as increasing returns to scale.: "Non-convexities in [both] production and consumption ... required mathematical tools that went beyond convexity, and further development had to await the invention of non-smooth calculus": For example, Clarke's differential calculus for Lipschitz continuous functions, which uses Rademacher's theorem and which is described by and ,Chapter 8 "Applications to economics", especially Section 8.5.3 "Enter nonconvexity" (and the remainder of the chapter), particularly page 495: according to .

Thus, the pseudotriangle is the region between the convex hulls of these three curves, and more generally any three mutually tangent convex sets form a pseudotriangle that lies between them. For algorithmic applications it is of particular interest to characterize pseudotriangles that are polygons. In a polygon, a vertex is convex if it spans an interior angle of less than π, and concave otherwise (in particular, we consider an angle of exactly π to be concave). Any polygon must have at least three convex angles, because the total exterior angle of a polygon is 2π, the convex angles contribute less than π each to this total, and the concave angles contribute zero or negative amounts.

If , then , and therefore . Since is convex, it then also contains the convex hull of and therefore also . Likewise, if , then , and by the same reasoning . Since is in every , it must also be in the intersection.

The protoconch is conical, small, acute, and consists of 2½ convex smooth and pinkish-brown whorls. The shell contains 8 to 10 whorls. The first very slowly, then rapidly increase. The whorls are straight or slightly convex.

Ammonites belonging to Microdactylites have small shells with evolute, compressed coiling. Flanks are slightly convex to convex and whorl section is suboval to subcircular. Ribbing is dense, while ribs can be both simple or bifurcating.Kovács, Z. (2014).

100x100px The set of all feasible solutions is an intersection of hyperspaces. Therefore, it is a convex polyhedron. If it is bounded, then it is a convex polytope. Each BFS corresponds to a vertex of this polytope.

In mathematics, specifically in order theory and functional analysis, a locally convex vector lattice (LCVL) is a topological vector lattice that is also a locally convex space. LCVLs are important in the theory of topological vector lattices.

The second edition also provides an improved bibliography. Topics that are important to the theory of convex polytopes but not well-covered in the book Convex Polytopes include Hilbert's third problem and the theory of Dehn invariants.

In 1967, however, Petty and Schneider showed that the answer is "no" for every n ≥ 3\. The solution of Shephard's problem requires Minkowski's first inequality for convex bodies and the notion of projection bodies of convex bodies.

The suture is simple. The shell has six whorls that are slightly convex, increasing with moderate rapidity. The last whorl is convex, not descending in front, somewhat attenuated at base. The columella is suboblique, sometimes nearly vertical.

The economics depends upon the following definitions and results from convex geometry.

The subderivative and subgradient are generalizations of the derivative to convex functions.

There are no non-convex regular polytopes in five dimensions or higher.

This observation also holds for any other convex polygon in the plane .

The varietal epithet convexa refers to the convex shape of the cap.

Hotelling made pioneering studies of non-convexity in economics. In economics, non-convexity refers to violations of the convexity assumptions of elementary economics. Basic economics textbooks concentrate on consumers with convex preferences and convex budget sets and on producers with convex production sets; for convex models, the predicted economic behavior is well understood. When convexity assumptions are violated, then many of the good properties of competitive markets need not hold: Thus, non-convexity is associated with market failures, where supply and demand differ or where market equilibria can be inefficient.

Geometric combinatorics is a branch of mathematics in general and combinatorics in particular. It includes a number of subareas such as polyhedral combinatorics (the study of faces of convex polyhedra), convex geometry (the study of convex sets, in particular combinatorics of their intersections), and discrete geometry, which in turn has many applications to computational geometry. Other important areas include metric geometry of polyhedra, such as the Cauchy theorem on rigidity of convex polytopes. The study of regular polytopes, Archimedean solids, and kissing numbers is also a part of geometric combinatorics.

A number of algorithms are known for the three- dimensional case, as well as for arbitrary dimensions.See David Mount's Lecture Notes, including Lecture 4 for recent developments, including Chan's algorithm. Chan's algorithm is used for dimensions 2 and 3, and Quickhull is used for computation of the convex hull in higher dimensions. For a finite set of points, the convex hull is a convex polyhedron in three dimensions, or in general a convex polytope for any number of dimensions, whose vertices are some of the points in the input set.

The Schlegel diagram of a convex polyhedron represents its vertices and edges as points and line segments in the Euclidean plane, forming a subdivision of an outer convex polygon into smaller convex polygons (a convex drawing of the graph of the polyhedron). It has no crossings, so every polyhedral graph is also a planar graph. Additionally, by Balinski's theorem, it is a 3-vertex-connected graph. According to Steinitz's theorem, these two graph-theoretic properties are enough to completely characterize the polyhedral graphs: they are exactly the 3-vertex-connected planar graphs.

In complex geometry and analysis, the notion of convexity and its generalizations play an important role in understanding function behavior. Examples of classes of functions with a rich structure are, in addition to the convex functions, the subharmonic functions and the plurisubharmonic functions. Geometrically, these classes of functions correspond to convex domains and pseudoconvex domains, but there are also other types of domains, for instance lineally convex domains which can be generalized using convex analysis. A great deal is already known about these domains, but there remain some fascinating, unsolved problems.

A polyomino is said to be convex if it is row and column convex. A polyomino is said to be directed if it contains a square, known as the root, such that every other square can be reached by movements of up or right one square, without leaving the polyomino. Directed polyominoes, column (or row) convex polyominoes, and convex polyominoes have been effectively enumerated by area n, as well as by some other parameters such as perimeter, using generating functions. A polyomino is equable if its area equals its perimeter.

The 1-skeleton of any k-dimensional convex polytope forms a k-vertex-connected graph (Balinski's theorem, ). As a partial converse, Steinitz's theorem states that any 3-vertex-connected planar graph forms the skeleton of a convex polyhedron.

The outer lip begins at the lower carina and is concavo-convex to the anterior notch. The columella is concavo-convex from behind forwards. Verco, J.C. 1906. Notes on South Australian marine Mollusca with descriptions of new species.

The aperture is rhombic. Its upper margin is regularly convex, not very thin, and thickened interiorly. It is separated from the basal margin by a groove, corresponding to the keel. The basal margin is convex, crenulated, thickened interiorly.

11 in . (James' theorem) Since norm-closed convex subsets in a Banach space are weakly closed,Theorem 2.5.16 in . it follows from the third property that closed bounded convex subsets of a reflexive space X are weakly compact.

In mathematics, specifically convex geometry, the normal fan of a convex polytope P is a polyhedral fan that is dual to P. Normal fans have applications to polyhedral combinatorics, linear programming, tropical geometry and other areas of mathematics.

A kite, as defined above, may be either convex or concave, but the word "kite" is often restricted to the convex variety. A concave kite is sometimes called a "dart" or "arrowhead", and is a type of pseudotriangle.

The base of the shell is white or faintly marked with rose around the outer border. The about five whorls are slightly convex. They are separated by subcanaliculate sutures. The outline of the spire is a little convex.

The intervals between the bands are longitudinally closely lineolate with blackish. The spire is elevated. The shell contains about 6 whorls. The upper ones are slightly convex, the last generally constricted and concave below the suture, then convex.

The more one cuts this shape, the lesser the area and the greater the perimeter. The convex hull remains the same. The Neuf-Brisach fortification perimeter is complicated. The shortest path around it is along its convex hull.

These concepts are named after Eduard Helly (1884-1943); Helly's theorem on convex sets, which gave rise to this notion, states that convex sets in Euclidean space of dimension n are a Helly family of order n + 1.

The sutures are impressed. The about 4½ whorls are convex, rounded, all over finely regularly spirally lirulate. The body whorl is rounded at the periphery, or very bluntly subangular. It is convex beneath and impressed around the umbilicus.

The cap of this mushroom is 2 to 10 cm across. It is convex and becomes broadly convex or almost flat. When fresh, this species is smooth and moist. It has a reddish-brown colour fading to cinnamon.

During eye evolution there existed an additional convex stadium of the retina (Figure 1, Left), presumably, which explains why today the left visual field is represented in the right hemisphere and vice versa. With the change from convex to concave retinas, the visual input changed sides on the retina, and was represented upside down in addition. During evolution existed a convex stadium of the retina.

We can drop the requirement for the algebra to be locally convex, but still a complete metric space. In this case, the underlying space may be called a Fréchet space or an F-space . If the requirement that the number of seminorms be countable is dropped, the algebra becomes locally convex (LC) or locally multiplicatively convex (LMC) (, ). A complete LMC algebra is called an Arens- Michael algebra .

Closed convex sets are convex sets that contain all their limit points. They can be characterised as the intersections of closed half-spaces (sets of point in space that lie on and to one side of a hyperplane). From what has just been said, it is clear that such intersections are convex, and they will also be closed sets. To prove the converse, i.e.

The following simple heuristic is often used as the first step in implementations of convex hull algorithms to improve their performance. It is based on the efficient convex hull algorithm by Selim Akl and G. T. Toussaint, 1978. The idea is to quickly exclude many points that would not be part of the convex hull anyway. This method is based on the following idea.

The red imported fire ant's promesonotum is strongly convex, whereas this feature is weakly convex in S. richteri. Upon examination, the base of the propodeum is elongated and straight in S. richteri, while convex and shorter in the red imported fire ant. It also has a wide postpetiole with either straight or diverging sides. The postpetiole in S. richteri is narrower with converging sides.

The extreme points of the closed unit disk in is the unit circle. Note that any open interval in has no extreme points while the extreme points of a non-degenerate closed interval are and . :Definition: A set is called convex if for any two points , contains the line segment . The smallest convex set containing is called the convex hull of and is denoted by .

The two diagonals of a convex quadrilateral are the line segments that connect opposite vertices. The two bimedians of a convex quadrilateral are the line segments that connect the midpoints of opposite sides. They intersect at the "vertex centroid" of the quadrilateral (see below). The four maltitudes of a convex quadrilateral are the perpendiculars to a side—through the midpoint of the opposite side.

The Erdős–Nagy theorem is a result in discrete geometry stating that a non- convex simple polygon can be made into a convex polygon by a finite sequence of flips. The flips are defined by taking a convex hull of a polygon and reflecting a pocket with respect to the boundary edge. The theorem is named after mathematicians Paul Erdős and Béla Szőkefalvi-Nagy.

Let P be a simple polygon or a rectifiable simple closed curve, and let X be any set enclosed by P. A geodesic between two points in P is a shortest path connecting those two points that stays entirely within P. A subset K of the points inside P is said to be relatively convex, geodesically convex, or P-convex if, for every two points of K, the geodesic between them in P stays within K. Then the relative convex hull of X can be defined as the intersection of all relatively convex sets containing X. Equivalently, the relative convex hull is the minimum-perimeter weakly simple polygon in P that encloses X. This was the original formulation of relative convex hulls, by . However this definition is complicated by the need to use weakly simple polygons (intuitively, polygons in which the polygon boundary can touch or overlap itself but not cross itself) instead of simple polygons when X is disconnected and its components are not all visible to each other.

In mathematics, concavification is the process of converting a non-concave function to a concave function. A related concept is convexification – converting a non-convex function to a convex function. It is especially important in economics and mathematical optimization.

If c is the center of symmetry of the smallest centrally-symmetric set containing a given convex body K, then the centrally- symmetric set itself is the convex hull of the union of K with its reflection across c.

The shell is oval, evenly convex, the two sides equally curved. The back of the shell is regularly convex, with little algal growth. The shell is not carinated at the row of holes. The spire is near the margin.

The base of the shell is slightly convex, and concentrically finely lirate. The sculpture is coarser than upon the upper surface. The large aperture is rounded-quadrate. The oblique columella is straightened and a little convex in the middle.

The Gram–Euler theorem similarly generalizes the alternating sum of internal angles \sum \varphi for convex polyhedra to higher-dimensional polytopes:M. A. Perles and G. C. Shephard. 1967. "Angle sums of convex polytopes". Math. Scandinavica, Vol 21, No 2.

The spire is high with its lateral outlines nearly straight . There are about 8 whorls, each one a trifle convex, the last angular at the periphery. The base of the shell is a little convex. The aperture is quadrate.

The convex hull of a figure may be visualized as the shape formed by a rubber band stretched around it. In the animated picture on the left, all the figures have the same convex hull; the big, first hexagon.

The length of the shell attains 4.8 mm, its width 2.1 mm. (Original description) The thin shell is translucent-white. It contains 6 whorls, including the protoconch of 2 smooth convex whorls. The whorls on the spire are convex.

The first two are polished, vitreous and convex. The others are rather convex, keeled above and flattened sloping downwards. The large body whorl is somewhat inflated and contracted towards the base. It contains about 12 strong furrows, longitudinally striated.

The hyoglossus depresses and retracts the tongue and makes the dorsum more convex.

Head with a cephalic knob. Canthal edges indistinct. Loreal region convex. Nostrils oval.

Unfortunately, this conflicts directly with Shapley's original definition of supermodular functions as "convex".

In 1994 V. D. Sedykh showed that every simple closed space curve which lies on the boundary of a convex body has four vertices. In 2015 Mohammad Ghomi generalized Sedykh's theorem to all curves which bound a locally convex disk.

An icosahedron. Geometric combinatorics is related to convex and discrete geometry, in particular polyhedral combinatorics. It asks, for example, how many faces of each dimension a convex polytope can have. Metric properties of polytopes play an important role as well, e.g.

If is TVS then the following are equivalent: 1. is locally convex and pseudometrizable. 2. has a countable neighborhood base at the origin consisting of convex sets. 3. The topology of is induced by a countable family of (continuous) seminorms. 4.

Each whorl is encircled by two more prominent, remote sulci. The shell contains 6 convex whorls, separated by deep sutures, and inflated above. The body whorl is subangulate, convex beneath, and contains numerous unequal concentric lirae. The aperture is rhomboidal.

The spire is obtuse and contains four convex whorls, separated by impressed sutures. The first whorl is narrow and slowly increasing. The body whorl is large, rather convex above, and rounded beneath. The large aperture is very oblique and subrotund.

A convex polytope, like any compact convex subset of Rn, is homeomorphic to a closed ball.Glen E. Bredon, Topology and Geometry, 1993, , p. 56. Let m denote the dimension of the polytope. If the polytope is full-dimensional, then m = n.

It should be distinguished from the kinetic convex hull, which studies similar problems for continuously moving points. Dynamic convex hull problems may be distinguished by the types of the input data and the allowed types of modification of the input data.

The sutures are impressed. The 7 whorls are convex, the last one rounded (or a trifle angled) around the lower part, slightly convex beneath. The rounded aperture is oblique. The outer lip is fluted w'thin, with a beveled opaque white submargin.

The shell is evenly horny- brownish, fragile and transparent, finely striated on the upper side, nearly smooth on the lower side. The shell has 5 weakly convex whorls with pronounced suture. The upper side is flattened. Lower side is convex.

The six whorls are slightly convex, the first buff, the remainder subangulate. The body whorl is dilated, slightly subangular in the middle, convex beneath and very finely decussated. The aperture is subovate and delicately sulcate within . The columella is arcuate.

Convex geometry is a relatively young mathematical discipline. Although the first known contributions to convex geometry date back to antiquity and can be traced in the works of Euclid and Archimedes, it became an independent branch of mathematics at the turn of the 20th century, mainly due to the works of Hermann Brunn and Hermann Minkowski in dimensions two and three. A big part of their results was soon generalized to spaces of higher dimensions, and in 1934 T. Bonnesen and W. Fenchel gave a comprehensive survey of convex geometry in Euclidean space Rn. Further development of convex geometry in the 20th century and its relations to numerous mathematical disciplines are summarized in the Handbook of convex geometry edited by P. M. Gruber and J. M. Wills.

There is also the question of whether any sufficiently large set of points in general position has an "empty" convex quadrilateral, pentagon, etc., that is, one that contains no other input point. The original solution to the happy ending problem can be adapted to show that any five points in general position have an empty convex quadrilateral, as shown in the illustration, and any ten points in general position have an empty convex pentagon.. However, there exist arbitrarily large sets of points in general position that contain no empty convex heptagon. For a long time the question of the existence of empty hexagons remained open, but and proved that every sufficiently large point set in general position contains a convex empty hexagon.

An early application of the convex layers was in robust statistics, as a way of identifying outliers and measuring the central tendency of a set of sample points. In this context, the number of convex layers surrounding a given point is called its convex hull peeling depth, and the convex layers themselves are the depth contours for this notion of data depth. Convex layers may be used as part of an efficient range reporting data structure for listing all of the points in a query half-plane. The points in the half-plane from each successive layer may be found by a binary search to find the most extreme point in the direction of the half-plane, and then searching sequentially from there.

A pocket of a non-convex simple polygon is a simple polygon bounded by a consecutive sequence of edges of the polygon together with a single edge of its convex hull that is not an edge of the polygon itself. Every convex hull edge that is not a polygon edge defines a pocket in this way. A flip of a pocket is obtained by reflecting the polygon edges that bound the pocket, across a reflection line containing the convex hull edge. Because the reflected pocket lies entirely within the reflected image of the convex hull, on the other side of this line, this operation cannot introduce any crossings, so the result of a flip is another simple polygon, with larger area.

In linear algebra, a convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients. A convex cone (light blue). Inside of it, the light red convex cone consists of all points αx + βy with α, β > 0, for the depicted x and y. The curves on the upper right symbolize that the regions are infinite in extent.

He illustrated three different concepts for incorporating reflective mirrors within his telescope model. Plan one consisted of a large, concave mirror directed towards the sun as to reflect light into a second, smaller, convex mirror. Cavalieri's second concept consisted of a main, truncated, paraboloid mirror and a second, convex mirror. His third option illustrated a strong resemblance to his previous concept, replacing the convex secondary lens with a concave lens.

It is ornamented with waved zigzag and acutely angular light and dark brown streaks. The 5½ whorls are slightly convex, moderately sloping, separated by simple, deep-channeled sutures. The spire shows only here and there very faint traces of obsolete spiral striae. The broad body whorl convex above with small erect slightly angular nodules in three rows, the two rounded at the periphery, depressly convex at the base.

In a scoliotic patient, the vertebral column experiences extension forces on the convex side and compression forces on the concave side. At the apical vertebra, average bone density for the concave cortical bone is higher than for the convex cortical bone, and cancellous bone density is higher for the concave side than for the convex side.Adam, C. & Askin, G. (2009). Lateral bone density variations in the scoliotic spine.

The key element of the Hahn-Banach theorem is fundamentally a result about the separation of two convex sets: }, and }. This sort of argument appears widely in convex geometry, optimization theory, and economics. Lemmas to this end derived from the original Hahn-Banach theorem are known as the Hahn–Banach separation theorems.Gabriel Nagy, Real Analysis lecture notes Often one assumes that the convex sets have additional structure; i.e.

There is a mailing list of Convex ex-employees, as well as frequent reunions.

Every Banach lattice, normed lattice, and Fréchet lattice is a locally convex vector lattice.

A linear-time algorithm that area-bisects two disjoint convex polygons is described by .

Convex had an unusually thorough interview process, which, for technical positions, included a grilling by a group of engineers. The extensive interview process carried over to other departments as well, where the key people who would be working with the prospective employee each interviewed the candidate, then met in roundtable to discuss whether or not to hire. Convex lasted longer than most minisupercomputer companies, and to celebrate this and more so to remind themselves of the difficulties of the market, Convex had a graveyard of former competitor companies on its property. Ex-employees of Convex jokingly refer to themselves as ex-cons.

Bauer's maximum principle is the following theorem in mathematical optimization: ::Any function that is convex and continuous, and defined on a set that is convex and compact, attains its maximum at some extreme point of that set. It is attributed to the German mathematician Heinz Bauer. Bauer's maximum principle immediately implies the analogue minimum principle: ::Any function that is concave and continuous, and defined on a set that is convex and compact, attains its minimum at some extreme point of that set. Since a linear function is simultaneously convex and concave, it satisfies both principles, i.e.

Convex polyhedra can be defined in three-dimensional hyperbolic space in the same way as in Euclidean space, as the convex hulls of finite sets of points. However, in hyperbolic space, it is also possible to consider ideal points as well as the points that lie within the space. An ideal polyhedron is the convex hull of a finite set of ideal points. Its faces are ideal polygons, but its edges are defined by entire hyperbolic lines rather than line segments, and its vertices (the ideal points of which it is the convex hull) do not lie within the hyperbolic space.

First, when V is not locally convex, the continuous dual may be equal to {0} and the map Ψ trivial. However, if V is Hausdorff and locally convex, the map Ψ is injective from V to the algebraic dual of the continuous dual, again as a consequence of the Hahn–Banach theorem.If V is locally convex but not Hausdorff, the kernel of Ψ is the smallest closed subspace containing {0}. Second, even in the locally convex setting, several natural vector space topologies can be defined on the continuous dual , so that the continuous double dual is not uniquely defined as a set.

Dykstra's algorithm is a method that computes a point in the intersection of convex sets, and is a variant of the alternating projection method (also called the projections onto convex sets method). In its simplest form, the method finds a point in the intersection of two convex sets by iteratively projecting onto each of the convex set; it differs from the alternating projection method in that there are intermediate steps. A parallel version of the algorithm was developed by Gaffke and Mathar. The method is named after Richard L. Dykstra who proposed it in the 1980s.

In this graph, triangle 1-2-5 is convex, but path 2-3-4 is not, because it does not include one of the two shortest paths from 2 to 4. In metric graph theory, a convex subgraph of an undirected graph G is a subgraph that includes every shortest path in G between two of its vertices. Thus, it is analogous to the definition of a convex set in geometry, a set that contains the line segment between every pair of its points. Convex subgraphs play an important role in the theory of partial cubes and median graphs.

In computational geometry, a number of algorithms are known for computing the convex hull for a finite set of points and for other geometric objects. Computing the convex hull means constructing an unambiguous, efficient representation of the required convex shape. Output representations that have been considered for convex hulls of point sets include a list of linear inequalities describing the facets of the hull, an undirected graph of facets and their adjacencies, or the full face lattice of the hull. In two dimensions, it may suffice more simply to list the points that are vertices, in their cyclic order around the hull.

In geometric modeling, one of the key properties of a Bézier curve is that it lies within the convex hull of its control points. This so-called "convex hull property" can be used, for instance, in quickly detecting intersections of these curves. In the geometry of boat and ship design, chain girth is a measurement of the size of a sailing vessel, defined using the convex hull of a cross-section of the hull of the vessel. It differs from the skin girth, the perimeter of the cross-section itself, except for boats and ships that have a convex hull.

Several authors have studied algorithms for constructing orthogonal convex hulls: ; ; ; . By the results of these authors, the orthogonal convex hull of points in the plane may be constructed in time , or possibly faster using integer searching data structures for points with integer coordinates.

A set of six points in the plane. The Classical Ortho-convex Hull is the point set it self. The Maximal Ortho-convex Hull of the point set of the top figure. It is formed by the point set and the colored area.

The mean width w of any compact shape S in two dimensions is p/π, where p is the perimeter of the convex hull of S. So w is the diameter of a circle with the same perimeter as the convex hull.

The length of the shell attains 25 mm, its diameter 7 mm. The fusiform, shining shell contains 12 whorls of which 2-3 are in the protoconch. These are smooth and convex. The subsequent whorls are concave at the top, then slightly convex.

The size of the shell varies between 14 mm and 54 mm. The violaceous shell is more or less marbled with chestnut, and more or less granular on the body whorl. The convex spire convex is conical and tuberculated. The aperture is violaceous.

Its largest diameter occupies about 2/5 of that of the shell. Its walls are finely striate, like the whole surface of the shell. The aperture is rhombic, its upper margin convex, the basal one nearly straight. The columellar margin is slightly convex.

"A convex distal posterior area ... is continuous with the posterior-most apical tooth and stays adjacent to a distal media area". This area is convex in P. carnifex. In P. carnifex, the third tooth's anterior edge is elongated, compared to in other species.

A function is convex if and only if its epigraph is a convex set. The epigraph of a real affine function g : Rn→R is a halfspace in Rn+1. A function is lower semicontinuous if and only if its epigraph is closed.

These are convex and spirally lirate, mostly eroded. The spire contains about 5 whorls, the body whorl rather large, and rounded at the periphery The base of the shell is convex. The suture is impressed. The subquadrangular aperture is iridescent and lirate within.

The larger keels are smooth or obsoletely granular. The five whorls are convex, the last obtusely angular. The base of the shell is flat or slightly convex and spirally lirate with equal lirae and spotted brown. The interstices are transversely neatly striate.

The cap is convex to plano-convex, measuring in diameter. The cap surface is dry, tomentose, or even somewhat felt-like, and the colour is brownish to yellowish-brown. The flesh turns bluish-green when injured. It has a mild odor and taste.

Lagrangian relaxation can also provide approximate solutions to difficult constrained problems. When the objective function is a convex function, then any local minimum will also be a global minimum. There exist efficient numerical techniques for minimizing convex functions, such as interior-point methods.

In mathematics, the Markov–Kakutani fixed-point theorem, named after Andrey Markov and Shizuo Kakutani, states that a commuting family of continuous affine self-mappings of a compact convex subset in a locally convex topological vector space has a common fixed point.

The elongated, turreted shell has a uniform white color. The spire is composed of eight whorls, separated by a closely channeled suture. The protoconch is embryonic, smooth and convex. The two following whorls are also convex, but decorated with striae slightly decurrent.

Colonies are generally smooth and low convex with shiny surfaces.ABIS Encyclopedia. “Genus Escherichia.” Regnum Prokaryote.

The cap initially has a convex shape before flattening; its diameter may reach up to .

A locally convex ultrabarrelled space is barrelled. Every ultrabarrelled space is a quasi-ultrabarrelled space.

Then in the 10th century Abu'l Wafa described the convex regular and quasiregular spherical polyhedra.

Capunti is a kind of short convex oval pasta resembling an open empty pea pod.

Receptacle convex to subulate, chaffy, the scarious chaff not embracing the smooth dorsally compressed achenes.

It is easy to construct an example for which the convex hull contains all input points, but after the insertion of a single point the convex hull becomes a triangle. And conversely, the deletion of a single point may produce the opposite drastic change of the size of the output. Therefore, if the convex hull is required to be reported in traditional way as a polygon, the lower bound for the worst- case computational complexity of the recomputation of the convex hull is \Omega(N), since this time is required for a mere reporting of the output. This lower bound is attainable, because several general-purpose convex hull algorithms run in linear time when input points are ordered in some way and logarithmic-time methods for dynamic maintenance of ordered data are well- known.

The information given by a CC system is sufficient to define a notion of a convex hull within a CC system. The convex hull is the set of ordered pairs pq of distinct points with the property that, for every third distinct point r, pqr belongs to the system. It forms a cycle, with the property that every three points of the cycle, in the same cyclic order, belong to the system. By adding points one at a time to a CC system, and maintaining the convex hull of the points added so far in its cyclic order using a binary search tree, it is possible to construct the convex hull in time O(n log n), matching the known time bounds for convex hull algorithms for Euclidean points.

A general expression for the shape of the path of the TDR convex hulls center of mass has yet to be derived. In order to maintain a smooth rolling motion the center of mass of a rolling body must maintain a constant height. All prime polysphericons, polycons, and platonicons and some of the TDR convex hulls share this property. Some of the TDR convex hulls, like the oloid, do not possess this property.

The converse, however, is not valid: there are positive semidefinite cones that do not admit any second-order cone representation. In fact, while any closed convex semialgebraic set in the plane can be written as a feasible region of a SOCP, it is known that there exist convex semialgebraic sets that are not representable by SDPs, that is, there exist convex semialgebraic sets that can not be written as a feasible region of a SDP.

If the points are random variables, then for a narrow but commonly encountered class of probability density functions, this throw-away pre-processing step will make a convex hull algorithm run in linear expected time, even if the worst-case complexity of the convex hull algorithm is quadratic in n.Luc Devroye and Godfried Toussaint, "A note on linear expected time algorithms for finding convex hulls," Computing, Vol. 26, 1981, pp. 361-366.

Consequently, an isomorphism between two given well-ordered sets will be unique. #Cauchy's theorem on geometry of convex polytopes states that a convex polytope is uniquely determined by the geometry of its faces and combinatorial adjacency rules. #Alexandrov's uniqueness theorem states that a convex polyhedron in three dimensions is uniquely determined by the metric space of geodesics on its surface. #Rigidity results in K-theory show isomorphisms between various algebraic K-theory groups.

In computational geometry, the potato peeling or convex skull problem is a problem of finding the convex polygon of the largest possible area that lies within a given non-convex polygon. It was posed independently by Goodman and Woo,.. As cited by . and solved in polynomial time by Chang and Yap.. The exponent of the polynomial time bound is high, but the same problem can also be accurately approximated in near-linear time..

In mathematics, an invariant convex cone is a closed convex cone in a Lie algebra of a connected Lie group that is invariant under inner automorphisms. The study of such cones was initiated by Ernest Vinberg and Bertram Kostant. For a simple Lie algebra, the existence of an invariant convex cone forces the Lie algebra to have a Hermitian structure, i.e. the maximal compact subgroup has center isomorphic to the circle group.

For the real symplectic group, the maximal and minimal cone coincide, so there is only one invariant convex cone. When one is properly contained in the other, there is a continuum of intermediate invariant convex cones. Invariant convex cones arise in the analysis of holomorphic semigroups in the complexification of the Lie group, first studied by Grigori Olshanskii. They are naturally associated with Hermitian symmetric spaces and their associated holomorphic discrete series.

In 1972, McMullen proposed the following problem:D. G. Larman (1972), "On Sets Projectively Equivalent to the Vertices of a Convex Polytope", Bulletin of the London Mathematical Society 4, pp.6-12 : Determine the largest number u(d) such that for any given u(d) points in general position in affine d-space Rd there is a projective transformation mapping these points into convex position (so they form the vertices of a convex polytope).

That last result also suggests that the Hahn-Banach theorem can often be used to locate a "nicer" topology in which to work. For example, many results in functional analysis assume that a space is Hausdorff or locally convex. However, suppose is a topological vector space, not necessarily Hausdorff or locally convex, but with a nonempty, proper, convex, open set . Then geometric Hahn-Banach implies that there is a hyperplane separating from any other point.

In particular, there must exist a nonzero functional on — that is, the continuous dual space is non-trivial. Considering with the weak topology induced by , then becomes locally convex; by the second bullet of geometric Hahn-Banach, the weak topology on this new space separates points. Thus with this weak topology becomes Hausdorff. This sometimes allows some results from locally convex topological vector spaces to be applied to non-Hausdorff and non-locally convex spaces.

The lower convex hull of points in the plane appears, in the form of a Newton polygon, in a letter from Isaac Newton to Henry Oldenburg in 1676.; see , page 336, and . The term "convex hull" itself appears as early as the work of , and the corresponding term in German appears earlier, for instance in Hans Rademacher's review of . Other terms, such as "convex envelope", were also used in this time frame.

The length of the shell attains 4.9 mm, its diameter 2.1 mm. (Original description) The solid shell has an elongate-oval shape. It consists of 6 whorls, including the pointed protoconch of 2½ smooth convex whorls. The whorls of the spire are slightly convex.

The columella is straightly convex. The outer lip thin, simple, crenulated, and toothed by the spirals. With a deep, narrow posterior sinus, bounded on one side by the sutural lira, and on the other by the nearest secondary lira. In profile the lip is convex.

The shell grows to a height of 5½ mm. The shell has a conical shape with rounded periphery and a slightly convex base, umbilicated, white, scarcely with a yellowish tinge. The 6½ whorls are convex. The nucleus is smooth, the subsequent whorl has concentric ribs.

The protoconch consists of 2½ smooth convex whorls. The shell contains 7 to 8 whorls, markedly convex, the base contracted. The suture is well marked. The aperture is slightly oblique, oval, angled above, produced into an oblique short and open siphonal canal, truncated below.

The conical, turreted spire is acuminate and somewhat scalariform. The about 7 whorls are very convex, spirally lirate, and radiately costate above. They are bicarinated at the periphery, and encircled by a deep canal. The convex base of the shell bears about 5 spiral lirae.

The small, smooth, bright shell has a turbinate shape. The outlines of the spire are convex, variously maculated with rose color and reddish brown. The four whorls are very convex, and rapidly increasing. The body whorl is produced anteriorly, separated by well impressed sutures.

The sutures are slightly impressed. The seven whorls are slightly convex. There are generally two or three stronger lirae near the middle or periphery, and this gives at times a slightly bicarinate outline to the body whorl. The flattened base is a little convex.

No spiral sculpture is visible. The spire is flat, the nucleus only being slightly raised. The protoconch consists of one smooth whorl, which is convex, and the first half very often slightly elevated. The teleoconch consists of two whorls, the last one flatly convex above.

In mathematics, Moreau's theorem is a result in convex analysis. It shows that sufficiently well-behaved convex functionals on Hilbert spaces are differentiable and the derivative is well-approximated by the so-called Yosida approximation, which is defined in terms of the resolvent operator.

The convex portion, its backside, faces south onto Hidalgo Street.Johnson, Philip."Transco Tower and Park ." Water cascades in vast channeled sheets from the narrower top rim of the circle to the wider base below, both on the concave side and on the convex side.

The cap has a diameter of 8 to 12.5 centimeters with a smooth, silky white surface. It is gray in the middle, and turns brownish over time. It is arched convex to flat arched (plano-convex) with a distinct hump. The edge is curved.

Octagonal zonogon Tessellation by irregular hexagonal zonogons Regular octagon tiled by squares and rhombi In geometry, a zonogon is a centrally symmetric convex polygon. Equivalently, it is a convex polygon whose sides can be grouped into parallel pairs with equal lengths and opposite orientations.

The spire is conic, turreted, of the same height as the aperture. The protoconch consists of about 2 whorls. The nucleus is smooth, convex, slightly lateral,the second whorl convex, and minutely reticulated. The 6½ subsequent whorls are distinctly shouldered, flatly rounded below the angle.

The length of the shell attains 6 mm, its diameter 2 mm. This species has no very marked distinctive character. The white oblong-ovate shell contains 7 whorls, of which 3 smooth and convex whorls in the protoconch. The subsequent whorls are slightly convex.

It has several shapes, viz. lid, bowls, shallow plate, pot with elongated body, basins and convex-sided pots. Rimless bowls are with either straight or slightly convex sides and pointed edge. A single fragment of the small bowl is thick-sided with bevelled-in edge.

The cap is convex to plano-convex, reaching dimensions of . The cap surface is sticky or tacky. The center of the cap is gray to brown with a gray edge. The white gills are closely crowded together and free from attachment to the stipe.

Like every regular convex polygon, the regular convex pentagon has a circumscribed circle. For a regular pentagon with successive vertices A, B, C, D, E, if P is any point on the circumcircle between points B and C, then PA + PD = PB + PC + PE.

If a polynomial is SOS-convex, then it is also convex. Since establishing whether a polynomial is SOS-convex amounts to solving a semidefinite programming problem, SOS-convexity can be used as a proxy to establishing if a polynomial is convex. In contrast, deciding if a generic polynomial of degree large than four is convex is a NP-hard problem. The first counterexample of a polynomial which is convex but not SOS-convex was constructed by Amir Ali Ahmadi and Pablo Parrilo in 2009. The polynomial is a homogeneous polynomial that is sum-of-squares and given by: > p(x)= 32 x_{1}^{8}+118 x_{1}^{6} x_{2}^{2}+40 x_{1}^{6} x_{3}^{2}+25 > x_{1}^{4} x_{2}^{4} -43 x_{1}^{4} x_{2}^{2} x_{3}^{2}-35 x_{1}^{4} > x_{3}^{4}+3 x_{1}^{2} x_{2}^{4} x_{3}^{2} -16 x_{1}^{2} x_{2}^{2} > x_{3}^{4}+24 x_{1}^{2} x_{3}^{6}+16 x_{2}^{8} +44 x_{2}^{6} x_{3}^{2}+70 > x_{2}^{4} x_{3}^{4}+60 x_{2}^{2} x_{3}^{6}+30 x_{3}^{8} In the same year, Grigoriy Blekherman proved in a non-constructive manner that there exist convex forms that is not representable as sum of squares.

The basal callus is irregularly convex, with two lumps or prominences, one behind the columellar lip.

The area of a convex regular polygon is the product of its semiperimeter and its apothem.

For every stellation of some convex polytope, there exists a dual faceting of the dual polytope.

Canthal edges are rounded. Loreal region is flat. Interorbital space is convex. Internarial space is flat.

There are various general formulas for the area K of a convex quadrilateral ABCD with sides .

Female is similar, but has a less convex pygidium and coarsely punctate head when comparing male.

Grain is oblong, obtusely trigonous, or concavo-convex, red-brown and rugulose on the ventral side.

They are moist, round, and convex with a butyraceous consistency and a slight gray-yellow color.

This fact may be used to prove minimization results for continuous convex functionals, in the same way that the Bolzano–Weierstrass theorem is used for continuous functions on . Among several variants, one simple statement is as follows: :If is a convex continuous function such that tends to when tends to , then admits a minimum at some point . This fact (and its various generalizations) are fundamental for direct methods in the calculus of variations. Minimization results for convex functionals are also a direct consequence of the slightly more abstract fact that closed bounded convex subsets in a Hilbert space are weakly compact, since is reflexive.

In geometry, Radon's theorem on convex sets, published by Johann Radon in 1921, states that any set of d + 2 points in Rd can be partitioned into two sets whose convex hulls intersect. A point in the intersection of these convex hulls is called a Radon point of the set. For example, in the case d = 2, any set of four points in the Euclidean plane can be partitioned in one of two ways. It may form a triple and a singleton, where the convex hull of the triple (a triangle) contains the singleton; alternatively, it may form two pairs of points that form the endpoints of two intersecting line segments.

The Slothouber–Graatsma puzzle is an example of a cube-packing puzzle using convex polycubes. More general puzzles involving the packing of convex rectangular blocks exist. The best known example is the Conway puzzle which asks for the packing of eighteen convex rectangular blocks into a 5 x 5 x 5 box. A harder convex rectangular block packing problem is to pack forty-one 1 x 2 x 4 blocks into a 7 x 7 x 7 box (thereby leaving 15 holes); the solution is analogous to the 5x5x5 case, and has three 1x1x5 cuboidal holes in mutually perpendicular directions covering all 7 slices.

A quasiconvex function that is not convex A function that is not quasiconvex: the set of points in the domain of the function for which the function values are below the dashed red line is the union of the two red intervals, which is not a convex set. The probability density function of the normal distribution is quasiconcave but not concave. The bivariate normal joint density is quasiconcave. In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form (-\infty,a) is a convex set.

For convex hulls in two or three dimensions, the complexity of the corresponding algorithms is usually estimated in terms of n, the number of input points, and h, the number of points on the convex hull, which may be significantly smaller than n. For higher-dimensional hulls, the number of faces of other dimensions may also come into the analysis. Graham scan can compute the convex hull of n points in the plane in time O(n\log n). For points in two and three dimensions, more complicated output-sensitive algorithms are known that compute the convex hull in time O(n\log h).

The length of the shell attains 5.7 mm, its diameter 2.1 mm. (Original description) The narrow shell has a fusiform shape. It contains 6 whorls, including the protoconch of 2 smooth convex whorls, with simple suture. The whorls of the spire are convex, with simple suture.

The size of the shell varies between 5 mm and 8.5 mm; its diameter 3 mm. The solid shell is very elongated and has a fusiform-subcylindrical shape. The spire contains 8 whorls; the first ones smooth and convex, the others slightly convex. The suture is appressed.

The Newton line is the line that connects the midpoints of the two diagonals in a convex quadrilateral that is not a parallelogram. The line segments connecting the midpoints of opposite sides of a convex quadrilateral intersect in a point that lies on the Newton line.

Non-integrally-convex set Let n = 2 and let X = { (0,0), (1,0), (2,0), (2,1) }. Its convex hull ch(X) contains, for example, the point y = (1.2, 0.5). The integer points nearby y are near(y) = {(1,0), (2,0), (1,1), (2,1) }. So X ∩ near(y) = {(1,0), (2,0), (2,1)}.

Architectonica perdix has a shell that reaches 65–83 mm in maximum dimension. This shell is low-spired and quite flattened, with a beaded surface. It has seven flatly convex whorls and the base of shell is slight convex in the centre. The sutures are finely incised.

The dynamic convex hull problem is a class of dynamic problems in computational geometry. The problem consists in the maintenance, i.e., keeping track, of the convex hull for input data undergoing a sequence of discrete changes, i.e., when input data elements may be inserted, deleted, or modified.

Its lateral outlines are concave on the upper part. The apex is acute. The sutures are scarcely discernible until the body whorl is reached. There are 8-9, flat whorls, but the last one is slightly convex above, obtusely angular at the periphery, and somewhat convex beneath.

Leucocelis feana can reach a length of about . These beetles have a deep glossy green body, more or less tinged with red in some specimens. Head is longitudinally convex and sparsely punctured. Elytra are quite convex and sculptured and they show ten rows of coarse arcuate punctures.

The ordinary spire whorls are separated by a canaliculate suture and are flattened posteriorly. The body whorl is convex in the anterior-third. The base of the shell is flatly convex and falsely umbilicated. The columella enters the umbilical depression, which is shallow and moderately narrow.

In convex geometry, two disjoint convex sets in n-dimensional Euclidean space are separated by a hyperplane, a result called the hyperplane separation theorem. In machine learning, hyperplanes are a key tool to create support vector machines for such tasks as computer vision and natural language processing.

For terms see gastropod shell The 3-4 x 1.5-2 mm. shell has 5-7 convex whorls which are slightly more convex than those of Hydrobia acuta neglecta. Smaller shells with 5 whorls are slightly less slender than those of Hydrobia neglecta. The suture is deep.

The points dualize to lines and the convex hull of the points dualizes to the upper and lower envelope of the set of lines. The vertices of the upper convex hull dualize to segments on the upper envelope. The vertices of the lower convex hull dualize to segments on the lower envelope. The range of slopes of the supporting lines of a point on the hull dualize to the x-interval of segment that point dualizes to.

In geometry, a developable roller is a convex solid whose surface consists of a single continuous developable face. While rolling on a plane, most developable rollers develop their entire surface so that all the points on the surface touch the rolling plane. All developable rollers have ruled surfaces. Four families of developable rollers have been described to date: the prime polysphericons, the convex hulls of the two disc rollers (TDR convex hulls), the polycons and the Platonicons.

This is true regardless of whether the Hilbert space is finite-dimensional or not. Geometrically, when the state is not expressible as a convex combination of other states, it is a pure state. The family of mixed states is a convex set and a state is pure if it is an extremal point of that set. It follows from the spectral theorem for compact self-adjoint operators that every mixed state is a countable convex combination of pure states.

The points dualize to lines and the convex hull of the points dualizes to the upper and lower envelope of the set of lines. The vertices of the upper convex hull dualize to segments on the upper envelope. The vertices of the lower convex hull dualize to segments on the lower envelope. The range of slopes of the supporting lines of a point on the hull dualize to the x-interval of segment that point dualizes to.

The consumer prefers the vector of goods (Qx, Qy) over other affordable vectors. At this optimal vector, the budget line supports the indifference curve I2. An optimal basket of goods occurs where the consumer's convex preference set is supported by the budget constraint, as shown in the diagram. If the preference set is convex, then the consumer's set of optimal decisions is a convex set, for example, a unique optimal basket (or even a line segment of optimal baskets).

Precision glass moulding can be used to produce a large variety of optical form elements such as spheres, aspheres, free-form elements and array-structures. Concerning the curvature of the lens elements, the following statements can be drawn: Acceptable lens shapes are most bi-convex, plano-convex and mild meniscus shapes. Not unacceptable but hard to mould are bi-concave lenses, steep meniscus lenses, and lenses with severe features (e.g. a bump on a convex surface).

According to most former employees, Convex was a very fun place at which to work. For some time, there were beer parties every Friday, and an annual Convex Beach Party (where a truck load of sand would be dumped on the parking lot to simulate a beach in Richardson, Texas). There was a fitness center and other recreational facilities on-site. Convex had a very clear and compelling mission statement: "The Fastest Computers Possible for Under $1M".

A convex mirror diagram showing the focus, focal length, centre of curvature, principal axis, etc. A convex mirror or diverging mirror is a curved mirror in which the reflective surface bulges towards the light source. Convex mirrors reflect light outwards, therefore they are not used to focus light. Such mirrors always form a virtual image, since the focal point (F) and the centre of curvature (2F) are both imaginary points "inside" the mirror, that cannot be reached.

For a rectilinear polygon which is half- orthogonally convex (i.e. only in the x direction), a minimum covering by orthogonally convex polygons can be found in time O(n^2), where n is the number of vertices of the polygon. The same is true for a covering by rectilinear star polygons. The number of orthogonally-convex components in a minimum covering can, in some cases, be found without finding the covering itself, in time O(n).

Let be a topological vector space (TVS). :Definition: A convex and balanced subset of a real or complex vector space is called a disk and it is said to be disked, absolutely convex, or convex balanced. :Definition: A barrel or a barrelled set in a TVS is a subset that is a closed absorbing disk. Note that the only topological requirement on a barrel is that it be a closed subset of the TVS; all other requirements (i.e.

They have slim body; head is longer than broad; snout acuminate; nostril lateral not visible from above; eye width is about the same length as distance from nostril to anterior corner of eye. Loreal area barely convex; upper lip fleshy; immediate lateral postorbital are convex; temporal area slightly convex; tympanum absent; dorsal postorbital crest developed but not prominent. Tibia long; foot shorter than tibia; relative length of toes: 1<2<3<5<4; metatarsal tubercles poorly developed.

A subset C of a vector space X is called a cone if for all real r > 0, rC ⊆ C. A cone is called pointed if it contains the origin. A cone C is convex if and only if C + C ⊆ C. The intersection of any non-empty family of cones (resp. convex cones) is again a cone (resp. convex cone); the same is true of the union of an increasing (under set inclusion) family of cones (resp.

Under the embedding, the nonempty compact convex sets are mapped to points in the range space. In Rådström's construction, this embedding is additive and positively homogeneous. Rådström's approach used ideas from the theory of topological semi-groups. Later, Lars Hörmander proved a variant of this theorem for locally convex topological vector spaces using the support function (of convex analysis); in Hörmander's approach, the range of the embedding was the Banach lattice L1, and the embedding was isotone.

Thus, the regular polyhedrathe (convex) Platonic solids and (star) Kepler–Poinsot polyhedraform dual pairs, where the regular tetrahedron is self-dual. The dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces. The dual of an isotoxal polyhedron (having equivalent edges) is also isotoxal. Duality is closely related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.

Geometric programs are not in general convex optimization problems, but they can be transformed to convex problems by a change of variables and a transformation of the objective and constraint functions. In particular, after performing the change of variables y_i = \log(x_i) and taking the log of the objective and constraint functions, the functions f_i, i.e., the posynomials, are transformed into log-sum-exp functions, which are convex, and the functions g_i, i.e., the monomials, become affine.

The FPP is also preserved by any retraction. According to Brouwer fixed point theorem every compact and convex subset of a Euclidean space has the FPP. More generally, according to the Schauder-Tychonoff fixed point theorem every compact and convex subset of a locally convex topological vector space has the FPP. Compactness alone does not imply the FPP and convexity is not even a topological property so it makes sense to ask how to topologically characterize the FPP.

Convex functions play an important role in many areas of mathematics. They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties. For instance, a strictly convex function on an open set has no more than one minimum. Even in infinite-dimensional spaces, under suitable additional hypotheses, convex functions continue to satisfy such properties and as a result, they are the most well-understood functionals in the calculus of variations.

The regular convex 4-polytopes are the four-dimensional analogs of the Platonic solids in three dimensions and the convex regular polygons in two dimensions. Five of them may be thought of as close analogs of the Platonic solids. One additional figure, the 24-cell, has no close three-dimensional equivalent. Each convex regular 4-polytope is bounded by a set of 3-dimensional cells which are all Platonic solids of the same type and size.

Of course, a contemporary zoo-keeper does not want to purchase half of an eagle and half of a lion. Thus, the zoo-keeper's preferences are non-convex: The zoo-keeper prefers having either animal to having any strictly convex combination of both. equilibrium: Consumers can jump between two separate allocations (of equal utility). When the consumer's preference set is non-convex, then (for some prices) the consumer's demand is not connected; A disconnected demand implies some discontinuous behavior by the consumer, as discussed by Harold Hotelling: > If indifference curves for purchases be thought of as possessing a wavy > character, convex to the origin in some regions and concave in others, we > are forced to the conclusion that it is only the portions convex to the > origin that can be regarded as possessing any importance, since the others > are essentially unobservable.

An illustration of Carathéodory's theorem for a square in R2 Carathéodory's theorem is a theorem in convex geometry. It states that if a point x of Rd lies in the convex hull of a set P, then x can be written as the convex combination of at most d + 1 points in P. Namely, there is a subset ′ of P consisting of d + 1 or fewer points such that x lies in the convex hull of ′. Equivalently, x lies in an r-simplex with vertices in P, where r \leq d. The smallest r that makes the last statement valid for each x in the convex hull of P is defined as the Carathéodory's number of P. Depending on the properties of P, upper bounds lower than the one provided by Carathéodory's theorem can be obtained.

As with the problem of convex hull construction, this problem has a long history of incorrect proofs.

The lateral meniscus gives off from its anterior convex margin a fasciculus which forms the transverse ligament.

A locally convex quasi-barreled space that is also a 𝜎-barrelled space is a barrelled space.

The lavabo has a convex base and faucett, and is decorated with florians and surmounted by pinnacles.

The specific epithet is derived from the Latin convexus (meaning convex) and refers to the forewing hump.

Females are similar to males, but slightly squatter and the forewing outer margin is slightly more convex.

Both front and back sides of the scale are slightly convex, while top is a rounded point.

Female is similar, but has a less convex pygidium and more acute clypeal teeth when comparing male.

A very small beetle with a convex body. Short antennae. Smooth frons and vertex. Frontal tubercles absent.

The "convex dimension" of an antimatroid is defined as the minimum number of chains needed to define the antimatroid, and Dilworth's theorem can be used to show that it equals the width of an associated partial order; this connection leads to a polynomial time algorithm for convex dimension .

In 2018, however, an international photometric survey, using archived photometric data from the Geneva Observatory as well from dedicated observations, modeled a far longer period of hours with an amplitude of magnitude (). The survey uses combines convex lightcurve inversion with a non-convex algorithm (SAGE) to derive their periods.

The cap is typically around 24 – 102 mm (2.4 – 10.2 cm) wide, is hemispheric at first then becoming broadly convex to plano-convex, occasionally also slightly depressed in center; white, pallid grayish-brown or grayish buff over disk in age, surface dull and tacky at first and becoming shiny.

It states that when the condition that be logarithmically convex (or "super-convex") is added, it uniquely determines for positive, real inputs. From there, the gamma function can be extended to all real and complex values (except the negative integers and zero) by using the unique analytic continuation of .

The forewings are clear light yellow with a very small fuscous basal patch, the edge is irregularly convex. There is a broad fuscous terminal fascia, occupying more than one-third of the wing, the edge nearly straight, slightly convex in the middle. The hindwings are grey whitish.Exotic Microlepidoptera.

Sharir et al. (2001). For points in three dimensions that are in convex position, that is, are the vertices of some convex polytope, the number of k-sets is Θ((n-k)k), which follows from arguments used for bounding the complexity of k-th order Voronoi diagrams.

Shape convex to low-convex above, flattened below. Whorls rounded, with shallow to very shallow sutures. Umbilicus moderately wide, symmetrical, deep, exposing upper whorls, usually slightly overlapped by reflected peristome. Mouth broadly oval, except where interrupted by penultimate whorl; last part of body whorl expanding, descending near mouth.

The apex is strongly convex with 3.5-4.5 convex and regularly increasing whorls. The last whorl is not inflated near the aperture and not descending. The aperture is slightly oblique and the umbilicus is wide. The animal is bluish grey with a lighter sole and bluish black upper tentacles.

The surface is very lightly obliquely striate, closely, densely finely spirally striate, generally with three strong carinae, one at periphery, the others above. The about 5 whorls are convex, those of the upper surface bicarinate. The convex body whorl is carinate or subcarinate. The oblique aperture is rounded- quadrangular.

The shell is obliquely striate, spirally lirate with 6 subequal lirae on the penultimate whorl. The body whorl is a little convex above, carinated in the middle, convex beneath and provided with 7–8 concentric, white-and-brown articulated lirae. The aperture is rhomboid. The columella is subtruncate below.

The cell has two flagella for locomotion. Reproduction is by simple binary fission. In lateral view D. acuminata cells are irregularly egg-shaped, dorsally convex and have large hypothecal plates with a more or less oval shape. The dorsal contour is always more strongly convex than the ventral one.

The size of the shell varies between 8 mm and 15 mm. The white, sublenticular shell is flattened convex above, more convex below. It contains oblique radiating riblets, interrupted by an obtuse peripheral rib The interstices of the riblets are finely spirally striated. The umbilicus has a moderate size.

The length of the shell varies between 4 mm and 8 mm. The smooth, shining shell has a globose-conic shape. The small shell is composed of four to five convex whorls, the two first very small, convex and depressed; the others very large. The sutures are linear.

The five whorls are very convex and smooth. Only under a strong lens very faint growth striae and microscopic punctuations may be observed. The sutures are well-marked and marginated. The body whorl is rounded, with a convex base and a small perforation, nearly concealed by the columella.

The distance between the circle centers equals the radius of the circles. One third of each circle's perimeter lies inside the convex hull, so the same shape may be also formed as the convex hull of the two remaining circular arcs each spanning an angle of 4π/3.

The scholar Ibn al-Haytham discussed concave and convex mirrors in both cylindrical and spherical geometries, carried out a number of experiments with mirrors, and solved the problem of finding the point on a convex mirror at which a ray coming from one point is reflected to another point.

The Schläfli symbol is a recursive description, starting with {p} for a p-sided regular polygon that is convex. For example, {3} is an equilateral triangle, {4} is a square, {5} a convex regular pentagon and so on. Regular star polygons are not convex, and their Schläfli symbols {p/q} contain irreducible fractions p/q, where p is the number of vertices, and q is their turning number. Equivalently, {p/q} is created from the vertices of {p}, connected every q.

This result is known as Hölder's theorem. A definite and generally applicable characterization of the gamma function was not given until 1922. Harald Bohr and Johannes Mollerup then proved what is known as the Bohr–Mollerup theorem: that the gamma function is the unique solution to the factorial recurrence relation that is positive and logarithmically convex for positive and whose value at 1 is 1 (a function is logarithmically convex if its logarithm is convex). Another characterisation is given by the Wielandt theorem.

Convex mirrors are used in some automated teller machines as a simple and handy security feature, allowing the users to see what is happening behind them. Similar devices are sold to be attached to ordinary computer monitors. Convex mirrors make everything seem smaller but cover a larger area of surveillance. Round convex mirrors called Oeil de Sorcière (French for "sorcerer's eye") were a popular luxury item from the 15th century onwards, shown in many depictions of interiors from that time.

This process is repeated until an optimal integer solution is found. Cutting-plane methods for general convex continuous optimization and variants are known under various names: Kelley's method, Kelley–Cheney–Goldstein method, and bundle methods. They are popularly used for non-differentiable convex minimization, where a convex objective function and its subgradient can be evaluated efficiently but usual gradient methods for differentiable optimization can not be used. This situation is most typical for the concave maximization of Lagrangian dual functions.

The task of computing the volume of a convex polytope has been studied in the field of computational geometry. The volume can be computed approximately, for instance, using the convex volume approximation technique, when having access to a membership oracle. As for exact computation, one obstacle is that, when given a representation of the convex polytope as an equation system of linear inequalities, the volume of the polytope may have a bit-length which is not polynomial in this representation.

Orthoscopic eyepiece diagram The 4-element orthographic eyepiece consists of a plano-convex singlet eye lens and a cemented convex-convex triplet field lens achromatic field lens. This gives the eyepiece a nearly perfect image quality and good eye relief, but a narrow apparent field of view — about 40°–45°. It was invented by Ernst Abbe in 1880. It is called "orthoscopic" or "orthographic" because of its low degree of distortion and is also sometimes called an "ortho" or "Abbe".

Note: if all agents' preferences are convex (as in theorem 1), then A(u) is obviously convex too. Moreover, if A(u) is singleton (as in theorem 2) then it is obviously convex too. Hence, Svensson's theorem is more general than both Varian's theorems. Theorem 4 (Diamantaras): If all agents' preferences are strongly monotone, and for every PE utility-profile u, the set A(u) is a contractible space (can be continuously shrunk to a point within that space), then PEEF allocations exist.

Convex Polyhedra is a book on the mathematics of convex polyhedra, written by Soviet mathematician Aleksandr Danilovich Aleksandrov, and originally published in Russian in 1950, under the title Выпуклые многогранники. It was translated into German by Wilhelm Süss as Konvexe Polyeder in 1958. An updated edition, translated into English by Nurlan S. Dairbekov, Semën Samsonovich Kutateladze and Alexei B. Sossinsky, with added material by Victor Zalgaller, L. A. Shor, and Yu. A. Volkov, was published as Convex Polyhedra by Springer- Verlag in 2005.

In combinatorial optimization and polyhedral combinatorics, central objects of study are the convex hulls of indicator vectors of solutions to a combinatorial problem. If the facets of these polytopes can be found, describing the polytopes as intersections of halfspaces, then algorithms based on linear programming can be used to find optimal solutions.; see especially remarks following Theorem 2.9. In multi- objective optimization, a different type of convex hull is also used, the convex hull of the weight vectors of solutions.

In it, Kepler set out the theoretical basis of double-convex converging lenses and double-concave diverging lenses—and how they are combined to produce a Galilean telescope—as well as the concepts of real vs. virtual images, upright vs. inverted images, and the effects of focal length on magnification and reduction. He also described an improved telescope—now known as the astronomical or Keplerian telescope—in which two convex lenses can produce higher magnification than Galileo's combination of convex and concave lenses.

Page 270: (Originally published as )Page 371: Such applications continued to motivate economists to study non-convex sets.

Page 270: (Originally published as )Page 371: Such applications continued to motivate economists to study non‑convex sets.

The top margin of the bone was convex at the front, followed by a concave region behind it.

Head convex and gently covered with yellow setae. Eyes small. Antennae submoniliform. Pronotum sparsely covered with yellow setae.

Clypeus slightly convex. No malar suture Antennae weakly setiform. Mesosoma not depressed. Pterostigma of fore wing rather narrow.

The shell is imperforate, conic and globular. Whorls are convex. The spire is short. The peristome is continuous.

The spire is conic. The apex is acute. The sutures are impressed. There are about seven, convex whorls.

For convex sets of distributions, Levi's works are instructive.Levi, I., 1990. Hard choices: Decision making under unresolved conflict.

In any locally convex TVS, the set of closed and bounded disks are a base of bounded set.

Therefore, the rotation distance on -node trees corresponds exactly to flip distance on triangulations of -sided convex polygons.

In common with many fells the western slopes are smooth and convex while the eastern side exhibits crags.

Hermann Minkowski proved, conversely, that every convex surface of constant girth is also a surface of constant width.

The chilidium is a convex plate often covering the cardinal process of the dorsal valve in the Protremata.

In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

It is guaranteed to be a vertex of the convex hull of the polygon. Alternatively, the vertex with the smallest Y-coordinate among the ones with the largest X-coordinates or the vertex with the smallest X-coordinate among the ones with the largest Y-coordinates (or any other of 8 "smallest, largest" X/Y combinations) will do as well. Once a vertex of the convex hull is choosen, one can then apply the formula using the previous and next vertices, even if those are not on the convex hull, as there can be no local concavity on this vertex. If the orientation of a convex polygon is sought, then, of course, any vertex may be picked.

Page 15 of: D. Leborgne Calcul différentiel et géométrie Puf (1982) . A slightly more general version is as follows:This version follows directly from the previous one because every convex compact subset of a Euclidean space is homeomorphic to a closed ball of the same dimension as the subset; see :;Convex compact set:Every continuous function from a convex compact subset K of a Euclidean space to K itself has a fixed point.V. & F. Bayart Point fixe, et théorèmes du point fixe on Bibmath.net. An even more general form is better known under a different name: :;Schauder fixed point theorem:Every continuous function from a convex compact subset K of a Banach space to K itself has a fixed point.

The constrictor and double constrictor are both extremely secure when tied tightly around convex objects with cord scaled for the task at hand. If binding around a not fully convex, or square-edged object, arrange the knot so the overhand knot portion is stretched across a convex portion, or a corner, with the riding turn squarely on top of it. In situations where the object leaves gaps under the knot and there are no corners, it is possible to finish the constrictor knot off with an additional overhand knot, in the fashion of a reef knot, to help stabilize it. Those recommendations aside, constrictor knots do function best on fully convex objects.

Given three points x_1, x_2, x_3 in a plane as shown in the figure, the point P is a convex combination of the three points, while Q is not. (Q is however an affine combination of the three points, as their affine hull is the entire plane.) In convex geometry, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. More formally, given a finite number of points x_1, x_2, \dots, x_n in a real vector space, a convex combination of these points is a point of the form :\alpha_1x_1+\alpha_2x_2+\cdots+\alpha_nx_n where the real numbers \alpha_i satisfy \alpha_i\ge 0 and \alpha_1+\alpha_2+\cdots+\alpha_n=1. As a particular example, every convex combination of two points lies on the line segment between the points.

The size of an adult shell varies between 8 mm and 1 mm. The dark yellowish-brown shell is elongate but has a narrow base. It contains eight whorls; the first two are convex, the others concave on top, convex below. The whorls are thus strongly constricted below the suture.

The length of the shell attains 7 mm, its diameter is 3 mm. (Original description) The short, solid shell is narrowly oval, with a blunt apex and a slightly contracted base. The protoconch consists of two smooth, slightly convex whorls. The four whorls of the spire are sloping scarcely convex.

Paraceraurus is a genus of trilobites that lived in the Ordovician period (485.4 to 443.4 Ma). Its remains have been found in China, Estonia, Sweden and North America.Paraceraurus in the Paleobiology Database These trilobites have a rounded and moderately convex cephalon. Glabella is convex or flattened, with a sub-rectangular outline.

The aperture is fulvous inside. The spire is conic, turriculate, very little higher than the aperture. The protoconch is papillate and consists of two smooth convex whorls. The shell contains 6 whorls, the last high in proportion, with a sloping, broad, and lightly excavated shoulder, slightly convex below the inconspicuous angle.

Scotiae could also occur in pairs, separated by a convex section called an astragal, or bead, narrower than a torus. Sometimes these sections were accompanied by still narrower convex sections, known as annulets or fillets.Hewson Clarke and John Dougall, The Cabinet of Arts, T. Kinnersley, London (1817), pp. 271, 272.

The base is convex, then contracted and ending in a short distally rounded beak. The suture is deep. The aperture is pyriform, broadly angled above, ending in a rather short almost straight siphonal canal, slightly turned to the left. The outer lip is imperfect, convex above, contracted near the base.

Graduate Texts in Mathematics, 171. Springer, Cham, 2016. xviii+499 pp. With Richard Bishop, he applied his submersion calculations to the geometry of warped products, in addition to studying the fundamental role of convex functions and convex sets in Riemannian geometry, and for the geometry of negative sectional curvature in particular.

In convex analysis, Popoviciu's inequality is an inequality about convex functions. It is similar to Jensen's inequality and was found in 1965 by Tiberiu Popoviciu,Popoviciu's paper has been published in Romanian language, but the interested reader can find his results in the review . Page 1 Page 2 a Romanian mathematician.

The base of the shell is smooth, tessellated around the irregularly convex, flesh-colored central callus. The shell contains six whorls, the last a little concave above, convex beneath. The subquadrate aperture is pearly inside. The circular callus is heaviest in front of the aperture and behind the columellar lip. .

A concave-convex cavity has one convex mirror with a negative radius of curvature. This design produces no intracavity focus of the beam, and is thus useful in very high-power lasers where the intensity of the intracavity light might be damaging to the intracavity medium if brought to a focus.

A large foramen is also present on the coracoid. The glenoid is broad and deep, slightly pointing to the outer lateral side. It has robust, convex crest-like borders. The supraglenoid thickness is developed in a convex crest-shaped form, it is divided across the top of the scapulocoracoid suture.

The shell is openly umbilicate (the umbilicus about one-fourth the total diameter), of a uniform pale brown tint, discoidal. The spire is convex but low. Suture is deeply impressed. The shell has 3 ½ whorls, that are convex, slowly increasing, the embryonic 1 ½ densely striate spirally, the rest radially costellate.

The cap is less than 23 mm across, with a convex shape and an incurved margin when young, expanding to broadly convex. The cap surface is smooth, often cracking with irregular fissures. The gills are gray to black. The stem is tall, 4 mm thick, and slightly swollen at the base.

Fruit bodies have convex to plano-convex caps measuring in diameter. The caps are dry with scales that can be either erect or flat on the surface. The colour is brown in the centre, becoming paler towards the edges. The flesh is white, and has a spermatic odour and mild taste.

This phenomenon is called the concave/convex, or simply up/down, ambiguity, and it confuses computer vision as well.

The cap, initially conical to convex in shape, flattens out with age and typically reaches diameters of up to .

They also have a smooth surface and a convex elevation. Its optimal temperature of growth is 37 degrees Celsius.

Such systems include LF-spaces. However, non-Hausdorff locally convex inductive limits do occur in natural questions of analysis.

The medial portion is the larger, and is slightly concave transversely; the lateral is convex above, slightly concave below.

The real absolute value function is a piecewise linear, convex function. Both the real and complex functions are idempotent.

Operations Research/Computer Science Interfaces Series, 41. Springer, New York, 2008. xx+383 pp. Murota, Kazuo Discrete convex analysis.

The elytrtra are oval, convex and covered throughout with small tubercles. Also the pronotum has a rugose, granular surface.

In convex analysis and mathematical optimization, the supporting functional is a generalization of the supporting hyperplane of a set.

Liberman's lemma is a theorem used in studying intrinsic geometry of convex surface. It is named after Joseph Liberman.

The labial surfaces of the plates are smooth, slightly convex both mesially and distally with a shallow concavity between.

It is natural to generalize orthogonal convexity to restricted-orientation convexity, in which a set is defined to be convex if all lines having one of a finite set of slopes must intersect in connected subsets; see e.g. , , or . In addition, the tight span of a finite metric space is closely related to the orthogonal convex hull. If a finite point set in the plane has a connected orthogonal convex hull, that hull is the tight span for the Manhattan distance on the point set.

The mucilaginous stipe is a characteristic feature The cap is shallowly convex to convex or irregularly convex, and with or without a shallow umbo, measuring up to in diameter and up to high. The cap margin is curved downward, sometimes slightly flared, and sometimes has translucent radial striations marking the positions of the gills underneath. The white flesh—thickest at the center of the cap—tapers gradually to the margin. The gills are broadly adnate (fused) to decurrent (running down the length of the stipe).

Initially, the Holmegaard bows were believed to have been made "backwards", that is with wood removed from the back and the belly made convex. This may be the result of a comparison with the English Longbow that has a flat back and a convex belly. Many successful replicas were made in this fashion even though working the back of the bow cuts the wood fibres and endangers the bow. Subsequent analysis suggested the back may have instead been convex with the flattened surface being the belly.

The Kutter (named after its inventor Anton Kutter) style uses a single concave primary, a convex secondary and a plano-convex lens between the secondary mirror and the focal plane, when needed (this is the case of the catadioptric Schiefspiegler). One variation of a multi-schiefspiegler uses a concave primary, convex secondary and a parabolic tertiary. One of the interesting aspects of some Schiefspieglers is that one of the mirrors can be involved in the light path twice — each light path reflects along a different meridional path.

The problem remains open, but over a sequence of papers researchers have tightened the gap between the known lower and upper bounds. In particular, constructed a (nonconvex) universal cover and showed that the minimum shape has area at most 0.260437; and gave weaker upper bounds. In the convex case, improved an upper bound to 0.270911861. used a min-max strategy for area of a convex set containing a segment, a triangle and a rectangle to show a lower bound of 0.232239 for a convex cover.

Detail of the convex mirror The small medallions set into the frame of the convex mirror at the back of the room show tiny scenes from the Passion of Christ and may represent God's promise of salvation for the figures reflected on the mirror's convex surface. Furthering the Memorial theory, all the scenes on the wife's side are of Christ's death and resurrection. Those on the husband's side concern Christ's life. The mirror itself may represent the eye of God observing the vows of the wedding.

Another definition of spatial network derives from the theory of space syntax. It can be notoriously difficult to decide what a spatial element should be in complex spaces involving large open areas or many interconnected paths. The originators of space syntax, Bill Hillier and Julienne Hanson use axial lines and convex spaces as the spatial elements. Loosely, an axial line is the 'longest line of sight and access' through open space, and a convex space the 'maximal convex polygon' that can be drawn in open space.

In geometry, the barycentric subdivision is a standard way of dividing an arbitrary convex polygon into triangles, a convex polyhedron into tetrahedra, or, in general, a convex polytope into simplices with the same dimension, by connecting the barycenters of their faces in a specific way. The name is also used in topology for a similar operation on cell complexes. The result is topologically equivalent to that of the geometric operation, but the parts have arbitrary shape and size. This is an example of a finite subdivision rule.

Petiole node in profile is high, taller than long, with anterior face weakly convex, dorsal face flat to weakly convex, and posterior faces weakly convex to weakly concave. Petiolar peduncle tapering broadly into petiolar node and approximately as long as petiolar node. Postpetiole in profile as tall or occasionally taller than petiole, approximately two times as tall as long; anterior face sloping evenly into dorsal face and junction of posterior face and dorsal face more angular. Dorsum of head covered with scattered to abundant weakly impressed foveae.

Convex Polytopes is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional convex polyhedra. It was written by Branko Grünbaum, with contributions from Victor Klee, Micha Perles, and G. C. Shephard, and published in 1967 by John Wiley & Sons. It went out of print in 1970. A second edition, prepared with the assistance of Volker Kaibel, Victor Klee, and Günter M. Ziegler, was published by Springer-Verlag in 2003, as volume 221 of their book series Graduate Texts in Mathematics.

A polygon that is already convex has no pockets. One can form a hierarchical description of any given polygon by constructing its hull and its pockets in this way and then recursively forming a hierarchy of the same type for each pocket. This structure, called a convex differences tree, can be constructed efficiently.

The colour of the shell is white. The spire is narrowly conical, measuring about 1½ times the height of the aperture. The protoconch consists of 1½ convex whorls, the nucleus is narrowly rounded and oblique. There are 4 to 5 subsequent whorls, regularly increasing, lightly convex and somewhat flattened below the suture.

The shell reaches a length of 7⅔ mm and a diameter of 3 mm. The ovate-fusiform shell is slighly pink with a reddish line around the body whorl. The shell contains eight whorls, including three smooth, convex whorls in the protoconch. The subsequent whorls are convex and somewhat shouldered above the middle.

Let P and Q be combinatorially equivalent 3-dimensional convex polytopes; that is, they are convex polytopes with isomorphic face lattices. Suppose further that each pair of corresponding faces from P and Q are congruent to each other, i.e. equal up to a rigid motion. Then P and Q are themselves congruent.

The shell contains seven whorls. The two white, opaque whorls in the protoconch are rounded and smooth,. The intermediates whorls are rather convex, angular on the first whorls in the upper part, becoming rounder on the penultimate whorl and rather convex on the body whorl. The whorls are crossed by longitudinal ribs.

These whorls are convex, separated by an undulating linear suture. The body whorl, which exceeds half the total height, presents, on the side opposite to the outer lip, a convex profile, more rapidly acuminated below the middle. The aperture is obliquely elongated, narrowly semi-oval. The peristome is continuous, white in the interior.

The length of the shell attains 7.5 mm, its diameter 2.5 mm. The turreted shell has an oblong shape. It contains 7 whorls, of which the first two are smooth and convex. The subsequent whorls are convex and contains 10-11 ribs that hardly stand out from the background and spiral lirae.

In mathematics, Tarski's plank problem is a question about coverings of convex regions in n-dimensional Euclidean space by "planks": regions between two hyperplanes. Tarski asked if the sum of the widths of the planks must be at least the minimum width of the convex region. The question was answered affirmatively by .

This idea was adapted by Marks et al. to the crystallographic phase problem. With a feasible set approach, constraints can be considered convex (highly convergent) or non- convex (weakly convergent). Imposing these constraints with the algorithm detailed earlier can converge towards unique or non-unique solutions, depending on the convexity of the constraints.

Another example is his theory of "Decentralization in NonConvex System" that was highlighted in Econometrica. "Decentralization in Non-Convex Systems" in Econometrica 1966 34:5S p. 123 - 125. The aim was to represent a two-level planning and decision making system within the traditional Edgeworth box diagram, but extended for non-convex analysis.

A subset C of X is a nonempty closed convex cone if and only if C++ = C∘∘ = C when C++ = (C+)+, where A+ denotes the positive dual cone of a set A. Or more generally, if C is a nonempty convex cone then the bipolar cone is given by :C∘∘ = cl(C).

The Pileus is 1.4–3.5 cm in diameter and conic to convex to broadly convex then becoming flat in age. It is not usually umbonate. The pileus is deep chestnut brown and hygrophanous, fading to yellowish brown or grayish white when dry. The surface is viscid when moist from the separable gelatinous pellicle.

In probability theory, a convex function applied to the expected value of a random variable is always bounded above by the expected value of the convex function of the random variable. This result, known as Jensen's inequality, can be used to deduce inequalities such as the arithmetic-geometric mean inequality and Hölder's inequality.

The crystalline lenses of fishes' eyes are extremely convex, almost spherical, and their refractive indices are the highest of all the animals. These properties enable proper focusing of the light rays and in turn proper image formation on the retina. This convex lens gives the name to the fisheye lens in photography.

Its dihedral angle is approximately 138.19°. Thus, just as hexagons have angles not less than 120° and cannot be used as the faces of a convex regular polyhedron because such a construction would not meet the requirement that at least three faces meet at a vertex and leave a positive defect for folding in three dimensions, icosahedra cannot be used as the cells of a convex regular polychoron because, similarly, at least three cells must meet at an edge and leave a positive defect for folding in four dimensions (in general for a convex polytope in n dimensions, at least three facets must meet at a peak and leave a positive defect for folding in n-space). However, when combined with suitable cells having smaller dihedral angles, icosahedra can be used as cells in semi-regular polychora (for example the snub 24-cell), just as hexagons can be used as faces in semi-regular polyhedra (for example the truncated icosahedron). Finally, non-convex polytopes do not carry the same strict requirements as convex polytopes, and icosahedra are indeed the cells of the icosahedral 120-cell, one of the ten non-convex regular polychora.

In mathematics, Choquet theory, named after Gustave Choquet, is an area of functional analysis and convex analysis concerned with measures which have support on the extreme points of a convex set C. Roughly speaking, every vector of C should appear as a weighted average of extreme points, a concept made more precise by generalizing the notion of weighted average from a convex combination to an integral taken over the set E of extreme points. Here C is a subset of a real vector space V, and the main thrust of the theory is to treat the cases where V is an infinite-dimensional (locally convex Hausdorff) topological vector space along lines similar to the finite-dimensional case. The main concerns of Gustave Choquet were in potential theory. Choquet theory has become a general paradigm, particularly for treating convex cones as determined by their extreme rays, and so for many different notions of positivity in mathematics.

The exterior border is slightly convex and very oblique. The hindwings are brownish cinereous.List Spec. Lepid. Insects Colln Br. Mus.

Every separoid can be represented with a family of convex sets in some Euclidean space and their separations by hyperplanes.

In some applications it is convenient to represent a convex polygon as an intersection of a set of half-planes.

The shell is whitish, sparsely maculated with dark brown. The three whorls are convex. The outer lip is ascending posteriorly.

The four whorls are convex. The suture is very deep. The aperture is circular. The thin peristome is slightly expanded.

If is locally convex and Hausdorff, then a disk is infrabornivorous if and only if it absorbs all compact disks.

The base is convex. The suture is impressed. The aperture is a little oblique, subcircular. The peristome is simple, straight.

The Bell numbers form a logarithmically convex sequence. Dividing them by the factorials, Bn/n!, gives a logarithmically concave sequence.

The entropic risk measure is a convex risk measure which is not coherent. It is related to the exponential utility.

For example, some blades may be flat-ground for much of the blade but be convex ground towards the edge.

The machine is not capable of sharpening drill bits in the standard profiles, or generating any convex or spiral profiles.

The periphery is rounded. The base of the shell is convex. The suture is impressed. The circular aperture is oblique.

Further, the floors have convex surface, or the edges of the floor is sloped so that cleaning would be easier.

For a finite set of points in the plane the lower bound on the computational complexity of finding the convex hull represented as a convex polygon is easily shown to be the same as for sorting using the following reduction. For the set x_1,\dots,x_n numbers to sort consider the set of points (x_1, x^2_1),\dots,(x_n, x^2_n) of points in the plane. Since they lie on a parabola, which is a convex curve it is easy to see that the vertices of the convex hull, when traversed along the boundary, produce the sorted order of the numbers x_1,\dots,x_n. Clearly, linear time is required for the described transformation of numbers into points and then extracting their sorted order.

Most barrier constructions are such that the fact that it covers the desired region is guaranteed. However, given an arbitrary barrier T, it would be desirable to confirm that it covers the desired area C. As a simple first pass, one can compare the convex hulls of C and T. T covers at most its convex hull, so if the convex hull of T does not strictly contain C, then it cannot possibly cover T. This provides a simple O(n log n) first-pass algorithm for verifying a barrier. If T consists of a single connected component, then it covers exactly its convex hull, and this algorithm is sufficient. However, If T contains more than one connected component, it may cover less.

Periodic tiling by the sphinx With pentagons that are not required to be convex, additional types of tiling are possible. An example is the sphinx tiling, an aperiodic tiling formed by a pentagonal rep- tile. The sphinx may also tile the plane periodically, by fitting two sphinx tiles together to form a parallelogram and then tiling the plane by translates of this parallelogram, a pattern that can be extended to any non-convex pentagon that has two consecutive angles adding to 2, thus satisfying the condition(s) of convex Type 1 above. It is possible to divide an equilateral triangle into three congruent non-convex pentagons, meeting at the center of the triangle, and to tile the plane with the resulting three-pentagon unit.

Each of these elements is defined by the geometry of the local boundary in different regions of the space map. Decomposition of a space map into a complete set of intersecting axial lines or overlapping convex spaces produces the axial map or overlapping convex map respectively. Algorithmic definitions of these maps exist, and this allows the mapping from an arbitrary shaped space map to a network amenable to graph mathematics to be carried out in a relatively well defined manner. Axial maps are used to analyse urban networks, where the system generally comprises linear segments, whereas convex maps are more often used to analyse building plans where space patterns are often more convexly articulated, however both convex and axial maps may be used in either situation.

Every convex kite has an inscribed circle; that is, there exists a circle that is tangent to all four sides. Therefore, every convex kite is a tangential quadrilateral. Additionally, if a convex kite is not a rhombus, there is another circle, outside the kite, tangent to the lines that pass through its four sides; therefore, every convex kite that is not a rhombus is an ex-tangential quadrilateral. For every concave kite there exist two circles tangent to all four (possibly extended) sides: one is interior to the kite and touches the two sides opposite from the concave angle, while the other circle is exterior to the kite and touches the kite on the two edges incident to the concave angle..

The above examples of mono-monostatic objects are necessarily inhomogeneous, that is, the density of their material varies across their body. The question of whether it is possible to construct a three-dimensional body which is mono-monostatic but also homogeneous and convex was raised by Russian mathematician Vladimir Arnold in 1995. The requirement of being convex is essential as it is trivial to construct a mono-monostatic non-convex body (an example would be a ball with a cavity inside it). Convex means that a straight line between any two points on a body lies inside the body, or, in other words, that the surface has no sunken regions but instead bulges outward (or is at least flat) at every point.

Suppose that X is a vector space over the field 𝔽 of real or complex numbers and ℬ is a vector bornology on X. Let 𝒩 denote all those subsets N of X that are convex, balanced, and bornivorous. Then 𝒩 forms a neighborhood basis at the origin for a locally convex TVS topology.

An example of generalized convexity is orthogonal convexity.Rawlins G.J.E. and Wood D, "Ortho-convexity and its generalizations", in: Computational Morphology, 137-152. Elsevier, 1988. A set in the Euclidean space is called orthogonally convex or ortho-convex, if any segment parallel to any of the coordinate axes connecting two points of lies totally within .

Simpler algorithms are possible for monotone polygons, star-shaped polygons, convex polygons and triangles. The triangle case can be solved easily by use of a barycentric coordinate system, parametric equation or dot product.Accurate point in triangle test "...the most famous methods to solve it" The dot product method extends naturally to any convex polygon.

Epinephelus sexfasciatus has a body with a standard length which is 2.7 to 3.2 times its depth. The dorsal profile of the head is convex and the intraorbital region is flat or slightly convex. The preopercle has 2 to 4 veryy enlarged serrations at its angle. The upper edge of the operculum is straight.

All regular simple polygons (a simple polygon is one that does not intersect itself anywhere) are convex. Those having the same number of sides are also similar. An n-sided convex regular polygon is denoted by its Schläfli symbol {n}. For n < 3, we have two degenerate cases: ; Monogon {1}: Degenerate in ordinary space.

The base of the shell scores by 5 or 6 narrow, spaced, concentric grooves that become stronger near the axis. The conic spire contains 6½ convex whorls. The body whorl is subangular at the periphery and convex beneath. The oblique aperture is brilliantly green inside, with a dusky submarginal band and a pale edge.

At low erosion rates increased stream incision may make gentle slopes to erode creating convex-up forms. Convex slopes around a stream can thus indirectly reflect accelerated crustal uplift. This is because accelerated incision may trigger accelerated erosion on the adjacent slopes creating slopes progressively steeper slopes that are in equilibrium with high erosion rates.

A similar analogue exists for finding the affine normal line at elliptic points of smooth surfaces in 3-space. This time one takes planes parallel to the tangent plane. These, for planes sufficiently close to the tangent plane, intersect the surface to make convex plane curves. Each convex plane curve has a centre of mass.

A Reuleaux triangle and its reflection enclosed by their smallest centrally symmetric convex superset, a regular hexagon In plane geometry the Estermann measure is a number defined for any bounded convex set describing how close to being centrally symmetric it is. It is the ratio of areas between the given set and its smallest centrally symmetric convex superset. It is one for a set that is centrally symmetric, and less than one for sets whose closure is not centrally symmetric. It is invariant under affine transformations of the plane.

Rosalina is a genus of foraminifera included in the rotaliid family Rosalinidae. Rosalina has a smooth plano-convex to concavo-convex trochospiral test in which the chambers are rapidly enlarging and all visible on the convex spiral side and subtriangular and strongly overlapping on the umbilical side, the final chamber taking up about one-third of the circumference. Sutures on the spiral side are depressed and oblique, curving back at the periphery. The umbilicus is open, partly covered by triangular umbilical flaps extending from each chamber of the final whorl.

So this test is not sufficient in general. The problem of determining exactly what regions any given forest T consisting of m connected components and n line segments actually covers, can be solved in Θ(m2n2) time. The basic procedure for doing this is simple: first, simplify each connected component by replacing it with its own convex hull. Then, for vertex p of each convex hull, perform a circular plane-sweep with a line centered at p, tracking when the line is or isn't piercing a convex hull (not including the point p itself).

If they do, they can be replaced by their combined convex hull without loss of generality. If after merging all overlapping hulls, a single barrier has resulted, then the more general algorithm need not be run; the coverage of a barrier is at most its convex hull, and we have just determined that its coverage is its convex hull. The merged hulls can be computed in O(nlog2n) time. Should more than one hull remain, the original algorithm can be run on the new simplified set of hulls, for a reduced running time.

The IBM ILOG CPLEX Optimizer solves integer programming problems, very large linear programming problems using either primal or dual variants of the simplex method or the barrier interior point method, convex and non-convex quadratic programming problems, and convex quadratically constrained problems (solved via second- order cone programming, or SOCP). The CPLEX Optimizer has a modeling layer called Concert that provides interfaces to the C++, C#, and Java languages. There is a Python language interface based on the C interface. Additionally, connectors to Microsoft Excel and MATLAB are provided.

A curve C is convex if and only if there are no three different points in C such that the tangents in these points are parallel. Proof: ⇒ If there are three parallel tangents, then one of them, say L, must be between the other two. This means that C lies on both sides of L, so it cannot be convex. ⇐ If C is not convex, then by definition there is point p on C such that the tangent line at p (call it L) has C on both sides of it.

For two convex polygons and in the plane with and vertices, their Minkowski sum is a convex polygon with at most + vertices and may be computed in time O ( + ) by a very simple procedure, which may be informally described as follows. Assume that the edges of a polygon are given and the direction, say, counterclockwise, along the polygon boundary. Then it is easily seen that these edges of the convex polygon are ordered by polar angle. Let us merge the ordered sequences of the directed edges from and into a single ordered sequence .

In the present context, the family of separable states is a convex set in the space of trace class operators. If ρ is an entangled state (thus lying outside the convex set), then by theorem above, there is a functional f separating ρ from the separable states. It is this functional f, or its identification as an operator, that we call an entanglement witness. There is more than one hyperplane separating a closed convex set from a point lying outside of it, so for an entangled state there is more than one entanglement witness.

The periphery of the body whorl is sometimes articulated with white, and the base of the shell is either unicolored dark, or finely dotted with white. The shell contains 10 whorls, the apical one or two convex and smooth, the following flat, finely spirally striate (about 14 striae on the penultimate whorl of a large specimen). The body whorl is convex at the periphery, angulated there in specimens not completely adult, and convex beneath, with 10-12 concentric lirulae there. The entire surface contains fine lines of growth.

For the sake of simplicity, the description below assumes that the points are in general position, i.e., no three points are collinear. The algorithm may be easily modified to deal with collinearity, including the choice whether it should report only extreme points (vertices of the convex hull) or all points that lie on the convex hull. Also, the complete implementation must deal with degenerate cases when the convex hull has only 1 or 2 vertices, as well as with the issues of limited arithmetic precision, both of computer computations and input data.

3D model of a snub disphenoid In geometry, the snub disphenoid, Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron is a three-dimensional convex polyhedron with twelve equilateral triangles as its faces. It is not a regular polyhedron because some vertices have four faces and others have five. It is a dodecahedron, one of the eight deltahedra (convex polyhedra with equilateral triangle faces) and one of the 92 Johnson solids (non-uniform convex polyhedra with regular faces). It can be thought of as a square antiprism where both squares are replaced with two equilateral triangles.

Even though the word "polygon" is used to describe this region, in general it can be any convex shape with curved edges. The support polygon is invariant under translations and rotations about the gravity vector (that is, if the contact points and friction cones were translated and rotated about the gravity vector, the support polygon is simply translated and rotated). If the friction cones are convex cones (as they typically are), the support polygon is always a convex region. It is also invariant to the mass of the object (provided it is nonzero).

In this context, a simplex in d-dimensional Euclidean space is the convex hull of d+1 points that do not all lie in a common hyperplane. For example, a 2-dimensional simplex is just a triangle (the convex hull of three points in the plane) and a 3-dimensional simplex is a tetrahedron (the convex of four points in three-dimensional space). The points that form the simplex in this way are called its vertices. An orthoscheme, also called a path simplex, is a special kind of simplex.

Its middle third is broad, slightly concave, and gives origin to the Abductor pollicis longus above, and the extensor pollicis brevis muscle below. Its lower third is broad, convex, and covered by the tendons of the muscles which subsequently run in the grooves on the lower end of the bone. The lateral surface (facies lateralis; external surface) is convex throughout its entire extent and is known as the convexity of the radius, curving outwards to be convex at the side. Its upper third gives insertion to the supinator muscle.

The convex hull of the concave polygon's vertices, and that of its edges, contains points that are exterior to the polygon.

Minkowski sums, specifically Minkowski differences, are often used alongside GJK algorithms to compute collision detection for convex hulls in physics engines.

Two (dicot) narrow lance shaped cotyledon with a tapered base, pointed tips, edges that convex to parallel and a hairless surface.

Lemaréchal's research also led to his work on (conjugate) subgradient methods and on bundle methods of descent for convex minimization problems.

The best known is the (convex, non-stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles.

A non-convex polytope may be self-intersecting; this class of polytopes include the star polytopes. Some regular polytopes are stars.

The fruit have 12-36 brown, flattened, elliptical, convex seeds that are 2.8-4 by 1.7-2 by 0.5-0.9 centimeters.

Astronaut 18, 1252-1260 With recent advancements in computing and optimization algorithms, convex programming is nearly as straightforward as linear programming.

Legs smooth, slender. Wings rather > broad; exterior border hardly festooned. Fore wings acute; costa straight; > exterior border slightly convex and oblique.

Plainview is described as having parallel or convex sides with a concave base. It is considered to be a Plano point.

The following whorl is spirally striate. The last two whorls are smooth. The six whorls are convex. The sutures are distinct.

If the consumer's utility function u(x) is locally nonsatiated and strictly convex, then h(p, u) = abla_p e(p, u).

Cephalic plate smooth and convex. Eight to ten ocelli present on each side. Tömösváry's organ moderately smaller. Tergites smooth, without wrinkles.

Various matrix completion algorithms has been proposed. These includes convex relaxation-based algorithm, gradient-based algorithm, and alternating minimization-based algorithm.

It is impossible to mill out even a convex cycloid or epicycloid, by the means and in the manner above described.

The shell contains 7 whorls, of which a little more than one forms a convex, smooth protoconch. The subsequent whorls are separated by a deep, waved suture. They are convex, angular and slightly excavated at their upper part. They show rather strong, rounded, oblique, axial ribs (11 on the body whorl), those behind the peristome stronger.

Archetypal analysis in statistics is an unsupervised learning method similar to cluster analysis and introduced by Adele Cutler and Leo Breiman in 1994. Rather than "typical" observations (cluster centers), it seeks extremal points in the multidimensional data, the "archetypes". The archetypes are convex combinations of observations chosen so that observations can be approximated by convex combinations of the archetypes.

This is a small to medium-sized agaric with a distinctively yellowish overall coloration. The cap has a diameter of up to and is yellow, often brownish towards the centre. The appearance of the cap may be convex to plano-convex. The volva is present as felty, floccose patches, 2–5 mm wide and up to 1 mm thick.

Net In four-dimensional geometry, a 16-cell is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called C16, hexadecachoron,N.W. Johnson: Geometries and Transformations, (2018) Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.

The body whorl is convex, but faintly contracted at the base. The suture is distinct, but little impressed. The aperture is lightly oblique, elongately oval, angled above, with a rudimentary, broad, and truncated siphonal canal below. The outer lip is convex, rather thin and sharp, smooth inside, with a very slight broad sinus below the suture.

The S- and X-class were Convex Exemplar SPP2000 supercomputers rebadged after HP's acquisition of Convex Computer in 1995. The S-class was a single-node SPP2000 with up to 16 processors, while the X-class name was used for multi-node configurations with up to 512 processors. These machines ran Convex's SPP-UX operating system.

Gnathothlibus heliodes is a moth of the family Sphingidae. It is known from Papua New Guinea and some adjacent islands. The outer margin of the forewing is straight or very slightly convex. The forewing upperside ground colour is light brown with a straight brown postmedian line, two slightly convex basal lines and two indistinct crenulated antemedian lines.

The continuous limit of a convex polygon is a parametrized convex curve in the plane. When the time parameter is suitably chosen, the continuous limit of the pentagram map is the classical Boussinesq equation. This equation is a classical example of an integrable partial differential equation. Here is a description of the geometric action of the Boussinesq equation.

Reflections in a convex mirror. The photographer is seen reflected at top right A curved mirror is a mirror with a curved reflecting surface. The surface may be either convex (bulging outward) or concave (recessed inward). Most curved mirrors have surfaces that are shaped like part of a sphere, but other shapes are sometimes used in optical devices.

Covering a polygon (which may contain holes) with convex polygons is NP-hard. There is an O(logn) approximation algorithm. Covering a polygon with convex polygons is NP-hard even when the target polygon is hole-free.. It is also APX-hard. The problem is NP-complete when the covering must not introduce new vertices (i.e.

The spire is conoidal with scarcely convex outlines. The about 6 whorls are somewhat convex and separated by well impressed sutures. The body whorl is large and deflected anteriorly. It bears 18 or 19 crowded, closely granose cinguli, of which the 1st, 3d, 5th, 7th, 9th and two upon the base are composed of alternate black and white granules.

The about 6 whorls are slightly convex, and spirally lirate. The body whorl is encircled by about 14 granose separated lirae, of which about 6 are on the upper surface, their interstices bearing spiral stripe. The body whorl is obtusely angular at the periphery, slightly convex beneath, a little descending anteriorly. The aperture is rounded-tetragonal.

Such dynamical system is called semi-dispersing billiard. If the walls are strictly convex, then the billiard is called dispersing. The naming is motivated by observation that a locally parallel beam of trajectories disperse after a collision with strictly convex part of a wall, but remain locally parallel after a collision with a flat section of a wall.

The cap is (2) 5–15 cm (23) across, convex with an incurved margin and expands to broadly convex to almost plane in age. The top is dry, fibrillose, and scaly, often with a blueish-green tinge when young. The color is variable, often with various bluish green, pink, or vinaceous patches. The cap is sometimes cracked in age.

Psilocybe serbica has no specific smell (somewhat raddish, but never farinaceous), taste is usually bitterish. It is a very variable species. Its cap is (1)2–4(5) cm in diameter and obtusely conical, later becoming campanulate or convex. It expands to broadly convex or plane in age and is incurved at first then plane or decurved with age.

Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols These properties apply to all regular polygons, whether convex or star. A regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points.

The earlier Greeks were interested primarily in the convex regular polyhedra, which came to be known as the Platonic solids. Pythagoras knew at least three of them, and Theaetetus (circa 417 B. C.) described all five. Eventually, Euclid described their construction in his Elements. Later, Archimedes expanded his study to the convex uniform polyhedra which now bear his name.

Quasi-bornological spaces where introduced by S. Iyahen in 1968. Every pseudometrizable TVS is quasi- bornological. A TVS in which every bornivorous set is a neighborhood of the origin is a quasi-bornological space. If is a quasi-bornological TVS then the finest locally convex topology on that is coarser than makes into a locally convex bornological space.

The body whorl is large and very convex. All these whorls are encircled by wide and distant ribs, slightly convex, numbering ten upon the body whorl. Others, more narrow, are placed alternately within the furrows, which are wide and very slightly striated. The surface of this shell is of a white color, slightly grayish, and sometimes rose-colored.

Although Blaschke sums of polytopes are used implicitly in the work of Hermann Minkowski, Blaschke sums are named for mathematician and Nazi Wilhelm Blaschke, who defined a corresponding operation for smooth convex sets. The Blaschke sum operation can be extended to arbitrary convex bodies, generalizing both the polytope and smooth cases, using measures on the Gauss map.

The painting depicts the young artist (then twenty one) in the middle of a room, distorted by the use of a convex mirror. The hand in the foreground is greatly elongated and distorted by the mirror. The work was painted on a specially-prepared convex panel in order to mimic the curve of the mirror used.

In small and medium workers, the head tends to have more elliptical sides. The head of small workers is wider out front than it is behind. In the major workers, the pronotum does not have any angular shoulders, nor does it have any sunken posteromedian area. The promesonotum is convex and the propodeum base is rounded and also convex.

The assumption of local convexity for the ambient space is necessary, because constructed a counter-example for the non-locally convex space where . Linearity is also needed, because the statement fails for weakly compact convex sets in CAT(0) spaces, as proved by . However, proved that the Krein–Milman theorem does hold for metrically compact CAT(0) spaces.

The theorem does not apply if one of the bodies is not convex. If one of A or B is not convex, then there are many possible counterexamples. For example, A and B could be concentric circles. A more subtle counterexample is one in which A and B are both closed but neither one is compact.

The outer lip is thickened by the last rib, angled above, then moderately convex, with a shallow sinus below the suture. The columella is vertical, straight, excavated on meeting the faintly convex parietal wall. The inner lip is narrow and thin. Henry Suter (1913): Manual of the New Zealand Mollusca; Government of New Zealand, Wellington, N.Z.

Reviewer Vasyl Gorkaviy recommends Convex Polyhedra to students and professional mathematicians as an introduction to the mathematics of convex polyhedra. He also writes that, over 50 years after its original publication, "it still remains of great interest for specialists", after being updated to include many new developments and to list new open problems in the area.

If P is an interior point in a convex quadrilateral ABCD, then :AP+BP+CP+DP\ge AC+BD. From this inequality it follows that the point inside a quadrilateral that minimizes the sum of distances to the vertices is the intersection of the diagonals. Hence that point is the Fermat point of a convex quadrilateral.

A Sorraia stallion with characteristic convex facial profile. The Sorraia breed stands between high, although some individuals are as small as The head tends to be large, the profile convex, and the ears long. The neck is slender and long, the withers high, and the croup slightly sloping. The legs are strong, with long pasterns and well-proportioned hooves.

The task of numerically computing the volume of objects is studied in the field of computational geometry in computer science, investigating efficient algorithms to perform this computation, approximately or exactly, for various types of objects. For instance, the convex volume approximation technique shows how to approximate the volume of any convex body using a membership oracle.

Its ovaries are 1.5-2.3 by 0.8-1.2 millimeters, covered in dense fine hairs, have a single chamber, convex backs, and flat faces. Each ovary has 7-8 ovules arranged in two rows. Its cone-shaped, broad stamens are 2.4 by 1.6 millimeters, concave on the back, and convex on their face. The stamen filaments are indistinct.

The siphon can be retracted completely into the shell. The two valves are triangular and convex, but the right valve is more convex than the left one. The siphon is protected by a horny sheath and it is provided with small tentacles at its end. The outer surface of the valves is covered with concentric growth lines.

The forewings are rounded at the tips, with three black points in the disk. The first point is found before the middle and the second is found behind the first, the third beyond the middle. The costa is convex towards the base and the exterior border is slightly convex, very obliquely. The hindwings are cinereous, tinged with aeneous.

In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformation, or Fenchel conjugate (after Adrien-Marie Legendre and Werner Fenchel). It allows in particular for a far reaching generalization of Lagrangian duality.

See, e.g., , page 520. By 1938, according to Lloyd Dines, the term "convex hull" had become standard; Dines adds that he finds the term unfortunate, because the colloquial meaning of the word "hull" would suggest that it refers to the surface of a shape, whereas the convex hull includes the interior and not just the surface.

The convex horseshoe bat (Rhinolophus convexus) is a species of bat in the family Rhinolophidae. It is found in Malaysia and Laos.

The cap is initially conic or parabolic, but expands somewhat in maturity to become convex, and typically reaches dimensions of up to .

The addition of a convex, long focus tertiary mirror leads to Leonard's Solano configuration. The Solano telescope doesn't contain any toric surfaces.

Another example is the convex set of probability distributions, as linear combinations preserve neither nonnegativity nor affinity (i.e., having total integral one).

The suture is impressed. The periphery is round and barely angulate. The base of the shell is convex. The umbilicus is moderate.

The increased strains on the convex side of the cupped wood panel causes further fracture in the ground layer as it dries.

The shell has a depressed-conical shape. It is widely umbilicate. The convex whorls are concentrically granose-lirate. The sutures are canaliculate.

The spire is conoidal. It contains about six convex whorls. The large body whorl is depressed-globose. The outer lip is simple.

Body is discoid to oval with the anterodorsal slightly convex. Head robust and wide. Snout blunt. Preanal spine and ectopterygoid teeth absent.

The Schmitt–Conway–Danzer tile, a convex polyhedron that tiles space, is not a stereohedron because all of its tilings are aperiodic.

Show slight sexual dimorphism. Male is about 10.7mm in length. Body elongate-oval and convex. A glossy black beetle with metallic sheen.

Cephalic plate smooth and convex. Six ocelli present on each side. Tömösváry's organ comparatively small and nearly rounded. Tergites smooth, without wrinkles.

Papers and Proceedings of the Royal Society of Tasmania 1882: 167–170 (described as Drillia woodsi) The thick and strong shell is elongately turreted, with a spire about 2½ times the length of the aperture, and consisting of a smooth convex translucent protoconch of about 1½ whorls, succeeded by about seven, gradually increasing nodose whorls. The apex is obtuse. The whorls are very slightly convex, with a well-marked suture, and a broad flat or very slightly convex area below the suture occupying a little less than half the breadth of the whorls. Below the sutural band, the whorls are more markedly convex owing to the presence of smooth oblique nodosities, which number from about ten to thirteen or fourteen to the whorl, usually with thirteen on the penultimate whorl.

More generally, a point in a convex set S is k-extreme if it lies in the interior of a k-dimensional convex set within S, but not a k+1-dimensional convex set within S. Thus, an extreme point is also a 0-extreme point. If S is a polytope, then the k-extreme points are exactly the interior points of the k-dimensional faces of S. More generally, for any convex set S, the k-extreme points are partitioned into k-dimensional open faces. The finite-dimensional Krein-Milman theorem, which is due to Minkowski, can be quickly proved using the concept of k-extreme points. If S is closed, bounded, and n-dimensional, and if p is a point in S, then p is k-extreme for some k < n.

Its vertex figure is a crossed quadrilateral. This model shares the name with the convex great rhombicosidodecahedron, also known as the truncated icosidodecahedron.

The convex hull of any finite set of points on the moment curve is a cyclic polytope.; , p. 101; , Lemma 5.4.2, p. 97.

There exist sequentially barrelled spaces that are not σ-quasi- barrelled. There exist quasi-complete locally convex TVSs that are not sequentially barrelled.

According to Lichti & Swihart (2011), kernel density methods provided, in many cases, less biased home-range area estimates compared to convex hull methods.

Darinka Dentcheva (Bulgarian: Даринка Денчева) is a Bulgarian-American mathematician, noted for her contributions to convex analysis, stochastic programming, and risk-averse optimization.

More general shapes of pieces have been studied, including: arbitrary convex polygons, spiral shapes, star polygons and monotone polygons. See for a survey.

Termen convex and oblique. Forewings deep purple with brownish irrorations (speckles). Markings are dark brown. A small whitish dot found on closing vein.

The spire is more or less elevated. The apex is obtuse. The sutures are impressed, sometimes subcanaliculate. The body whorl is convex beneath.

Silva worked in analytic functionals, the theory of distributions, vector- valued distributions, ultradistributions, the operational calculus, and differential calculus in locally convex spaces.

Snout to vent length 12–13.4 mm. Body stout in shape. Head is laterally convex. In lateral and dorsal view snout oval shaped.

Species range in shape from oval to very convex. The scutellum is concealed, the elytra have eight striae, and the clypeus is bidentate.

The fruiting forms (apothecia) are flat to slightly convex, and deep red-brown. It is in the Koerberia genus in the Placynthiaceae family.

The solid, white shell is rimate. it is subpellucid, smooth, and shining, white. The four whorls are convex. The thick outer lip simple.

Labial palpi convex, dorsal edge rounded. Forewing dark grayish brown. Postmedial line dark and conspicuous. Bright ocherous reniform stigma is narrow and triangular.

Finding an efficient kinetic data structure for maintaining the convex hull of moving points in dimensions higher than 2 is an open problem.

Fesnels Biprism complete set up this means optical bench type set up consisting of Fresnels biprism double convex lens, sodium light source etc.

The under or concave side of the voussoirs is called the intrados, and the upper or convex side the extrados of the arch.

The Szilassi polyhedron, a non-convex polyhedral realization of the Heawood graph with the topology of a torus In any dimension higher than three, the algorithmic Steinitz problem (given a lattice, determine whether it is the face lattice of a convex polytope) is complete for the existential theory of the reals by Richter-Gebert's universality theorem. However, because a given graph may correspond to more than one face lattice, it is difficult to extend this completeness result to the problem of recognizing the graphs of 4-polytopes, and this problem's complexity remains open. Researchers have also found graph-theoretic characterizations of the graphs of certain special classes of three-dimensional non-convex polyhedra.. and four-dimensional convex polytopes.... However, in both cases, the general problem remains unsolved. Indeed, even the problem of determining which complete graphs are the graphs of non-convex polyhedra (other than K4 for the tetrahedron and K7 for the Császár polyhedron) remains unsolved.. László Lovász has shown a correspondence between polyhedral representations of graphs and matrices realizing the Colin de Verdière graph invariants of the same graphs..

In this more general context, the convex hull of a set S is the intersection of the family members that contain S, and the Radon number of a space is the smallest r such that any r points have two subsets whose convex hulls intersect. Similarly, one can define the Helly number h and the Carathéodory number c by analogy to their definitions for convex sets in Euclidean spaces, and it can be shown that these numbers satisfy the inequalities h < r ≤ ch + 1.. In an arbitrary undirected graph, one may define a convex set to be a set of vertices that includes every induced path connecting a pair of vertices in the set. With this definition, every set of ω + 1 vertices in the graph can be partitioned into two subsets whose convex hulls intersect, and ω + 1 is the minimum number for which this is possible, where ω is the clique number of the given graph.. For related results involving shortest paths instead of induced paths see and .

To rotate, a light mill does not have to be coated with different colors across each vane. In 2009, researchers at the University of Texas, Austin created a monocolored light mill which has four curved vanes; each vane forms a convex and a concave surface. The light mill is uniformly coated by gold nanocrystals, which are a strong light absorber. Upon exposure, due to geometric effect, the convex side of the vane receives more photon energy than the concave side does, and subsequently the gas molecules receive more heat from the convex side than from the concave side.

It can be proven by mathematical induction (as Steinitz did), by finding the minimum-energy state of a two-dimensional spring system and lifting the result into three dimensions, or by using the circle packing theorem. Several extensions of the theorem are known, in which the polyhedron that realizes a given graph has additional constraints; for instance, every polyhedral graph is the graph of a convex polyhedron with integer coordinates, or the graph of a convex polyhedron all of whose edges are tangent to a common midsphere. In higher dimensions, the problem of characterizing the graphs of convex polytopes remains open.

In measure and probability theory in mathematics, a convex measure is a probability measure that -- loosely put -- does not assign more mass to any intermediate set "between" two measurable sets A and B than it does to A or B individually. There are multiple ways in which the comparison between the probabilities of A and B and the intermediate set can be made, leading to multiple definitions of convexity, such as log-concavity, harmonic convexity, and so on. The mathematician Christer Borell was a pioneer of the detailed study of convex measures on locally convex spaces in the 1970s.

Stochastic Network Optimization with Application to Communication and Queueing Systems, Morgan & Claypool, 2010. This is done by defining an appropriate set of virtual queues. It can also be used to produce time averaged solutions to convex optimization problems. M. J. Neely, "[Distributed and Secure Computation of Convex Programs over a Network of Connected Processors Distributed and Secure Computation of Convex Programs over a Network of Connected Processors]," DCDIS Conf, Guelph, Ontario, July 2005 S. Supittayapornpong and M. J. Neely, "Quality of Information Maximization for Wireless Networks via a Fully Separable Quadratic Policy," arXiv:1211.6162v2, Nov. 2012.

So the family of separable states is the closed convex hull of pure product states. We will make use of the following variant of Hahn–Banach theorem: Theorem Let S_1 and S_2 be disjoint convex closed sets in a real Banach space and one of them is compact, then there exists a bounded functional f separating the two sets. This is a generalization of the fact that, in real Euclidean space, given a convex set and a point outside, there always exists an affine subspace separating the two. The affine subspace manifests itself as the functional f.

The skeleton of any convex polyhedron is a planar graph, and the skeleton of any k-dimensional convex polytope is a k-connected graph. Conversely, Steinitz's theorem states that any 3-connected planar graph is the skeleton of a convex polyhedron; for this reason, this class of graphs is also known as the polyhedral graphs. A Euclidean graph is a graph in which the vertices represent points in the plane, and the edges are assigned lengths equal to the Euclidean distance between those points. The Euclidean minimum spanning tree is the minimum spanning tree of a Euclidean complete graph.

He visited the Institute for Advanced Study (IAS) in Princeton from 1948 to 1950, where he co-organized a seminar on convexity. Together with Olof Hanner, who, like Rådström, would earn his Ph.D. from Stockholm University in 1952, he improved Werner Fenchel's version of Carathéodory's lemma. In the 1950s, he obtained important results on convex sets. He proved the Rådström embedding theorem, which implies that the collection of all nonempty compact convex subsets of a normed real vector- space (endowed with the Hausdorff distance) can be isometrically embedded as a convex cone in a normed real vector-space.

The invariant convex cone generated by a generator of the Lie algebra of the center is closed and is the minimal invariant convex cone (up to a sign). The dual cone with respect to the Killing form is the maximal invariant convex cone. Any intermediate cone is uniquely determined by its intersection with the Lie algebra of a maximal torus in a maximal compact subgroup. The intersection is invariant under the Weyl group of the maximal torus and the orbit of every point in the interior of the cone intersects the interior of the Weyl group invariant cone.

In geometry Mukhopadhyaya's theorem may refer to one of several closely related theorems about the number of vertices of a curve due to . One version, called the Four-vertex theorem, states that a simple convex curve in the plane has at least 4 vertices, and another version states that a simple convex curve in the affine plane has at least 6 affine vertices.

The outer lip is flat at the shoulder, angulated at the keel, scarcely convex below this. The edge projects as a thin sharp lamina beyond the last longitudinal rib, which serves as a varix from the point of the shell to the keel. The edge is hardly convex, and scarcely forms a shoulder above. The sinus is merely a small rounded hollow.

Convex polygons are star shaped, and a convex polygon coincides with its own kernel. Regular star polygons are star-shaped, with their center always in the kernel. Antiparallelograms and self-intersecting Lemoine hexagons are star- shaped, with the kernel consisting of a single point. Visibility polygons are star-shaped as every point within them must be visible to the center by definition.

The first two are smooth, rounded and forming a papillary apex. The next two whorls are slightly convex and nearly smooth. The rest have the upper half slightly concave, with a rounded slightly tubercular ridge just below the suture. The lower are rather convex, furnished with a row of oblong nodules, or short stout costae (9 on the penultimate whorl).

Sidespot goatfish Parupeneus pleurostigma can reach a length of . They have nine dorsal soft rays, seven anal soft ray and sixteen pectoral rays. Snout is slightly convex and the margin of caudal-fin lobes is straight to slightly convex. Body color range from pinkish to yellowish gray, with a broad black spot on lateral line, sometimes followed by a large pinkish spot.

Discrete and Computational Geometry, Vol. 16, pp.361-368\. 1996. named after Timothy M. Chan, is an optimal output- sensitive algorithm to compute the convex hull of a set P of n points, in 2- or 3-dimensional space. The algorithm takes O(n \log h) time, where h is the number of vertices of the output (the convex hull).

The surface of the shell is polished, shining, with a few shallow spiral sulci on the upper surface, generally not more than 4, frequently obsolete. The shell contains about six whorls, each with a prominent, convex margin bordering the deeply impressed suture, below this margin concave. The body whorl is rounded at the periphery and convex beneath. The aperture is subquadrate.

Epinephelus bruneus has an elongate body which has a standard length which is 3.0 to 3.6 times its depth. The dorsal profile of the head between the eyes is convex. The preopercle has an angle where the serrations are notably enlarged. There is a small spine on the upper edge of the gill cover and this upper edge is convex.

HP V-Class computer. In 1995, Hewlett-Packard bought Convex. HP sold Convex Exemplar machines under the S-Class (MP) and X-Class (CC-NUMA) titles, and later incorporated some of Exemplar's technology into the V-Class machine, which was released running the HP-UX 11.0 release instead of the SPP-UX version which was sold with the S- and X-Class products.

The cap is in diameter, conic when young, bell-shaped to convex or plano-convex when mature, and umbonate. The cap margin is curved downward, even or slightly eroded. The red-orange cap surface is dry to moist, and wrinkled towards the margin but smooths out as it approaches the center. Sometimes there is white-yellowish flesh underneath the cap cuticle.

A convex set in light blue, and its extreme points in red. In mathematics, an extreme point of a convex set S in a real vector space is a point in S which does not lie in any open line segment joining two points of S. In linear programming problems, an extreme point is also called vertex or corner point of S.

Members of the Plectronocerida are characterized as follows. Shells are generally small, some even tiny, laterally compressed, curved (cyrtochonic) or straight (orthoconic). Most cyrtoconic forms are endogastric, with the ventral side longitudinally concave, or the dorsal side more longitudinally convex. A few, the two known genera in Balkoceratidae are exogastrically curved, with the ventral side convex and dorsal side concave.

It contains 8 whorls. The apical ones are smooth rounded oblique. The rest are deeply and smoothly concave at the top, then slightly convex, furnished with numerous oblique rounded smooth close-set ribs, the ribs terminating in a well-defined angle at the top. The body whorl is about equal in length to the spire, slightly convex above and tapering below.

In his doctoral dissertation and numerous later publications, Rockafellar developed a general duality theory based on convex conjugate functions that centers on embedding a problem within a family of problems obtained by a perturbation of parameters. This encapsulates linear programming duality and Lagrangian duality, and extends to general convex problems as well as nonconvex ones, especially when combined with an augmentation.

Gymnopilus maritimus mushrooms have a cap of between in width that is convex to flattened-convex in shape. There is sometimes a broad umbo, and in older specimens, the cap is depressed in the centre. The margin of the cap is somewhat wavey. The cap surface is dry and dull, coloured red to red-orange, and yellow towards the margin.

The concept of a triangulation may also be generalized somewhat to subdivisions into shapes related to triangles. In particular, a pseudotriangulation of a point set is a partition of the convex hull of the points into pseudotriangles, polygons that like triangles have exactly three convex vertices. As in point set triangulations, pseudotriangulations are required to have their vertices at the given input points.

The orange curves apply to Abby, and are convex as seen from the top right. Moving up and to the right increases Octavio’s allocation and puts him onto a more desirable indifference curve while placing Abby onto a less desirable one. Convex indifference curves are considered to be the usual case. They correspond to diminishing returns for each good relative to the other.

Members of the breed have heads of medium length, with a straight or slightly convex profile. Ultra convex and concave profiles are discouraged in the breed, and are penalized in breed shows. Necks are long and broad, running to well-defined withers and a massive chest. They have a short back and broad, strong hindquarters with a well-rounded croup.

In mathematics, particularly, in asymptotic convex geometry, Milman's reverse Brunn–Minkowski inequality is a result due to Vitali Milman that provides a reverse inequality to the famous Brunn-Minkowski inequality for convex bodies in n-dimensional Euclidean space Rn. Namely, it bounds the volume of the Minkowski sum of two bodies from above in terms of the volumes of the bodies.

Self-intersecting polyhedral Klein bottle with quadrilateral faces Some polyhedra have two distinct sides to their surface. For example, the inside and outside of a convex polyhedron paper model can each be given a different colour (although the inside colour will be hidden from view). These polyhedra are orientable. The same is true for non-convex polyhedra without self-crossings.

The dorsal profile is straight to slightly convex; the ventral profile is convex at the abdomen and straight posteriorly. The caudal peduncle depth is approximately equal to its length. The head in profile is acutely triangular overall with a bluntly rounded snout. The eyes are placed on the sides of the head and are visible from above, but not from below.

Every convex polytope may be dissected into simplexes. Therefore, if Hadwiger's conjecture is true, every convex polytope would also have a dissection into orthoschemes. A related result is that every orthoscheme can itself be dissected into d or d+1 smaller orthoschemes. Therefore, for simplexes that can be partitioned into orthoschemes, their dissections can have arbitrarily large numbers of orthoschemes.

If is convex and the origin to belong to the algebraic interior of , then is a non-negative sublinear functional on , which implies in particular that it is subadditive and positive homogeneous. In order for to be a seminorm, it suffices for to be a disk (i.e. convex and balanced) and absorbing in , which are the most common assumption placed on .

The theorem was conjectured and proven for special cases, such as Banach spaces, by Juliusz Schauder in 1930. His conjecture for the general case was published in the Scottish book. In 1934, Tychonoff proved the theorem for the case when K is a compact convex subset of a locally convex space. This version is known as the Schauder–Tychonoff fixed point theorem.

For any measurable space, the finite measures form a convex cone in the Banach space of signed measures with the total variation norm. Important subsets of the finite measures are the sub-probability measures, that form a convex subset, and the probability measures, which are the intersection of the unit sphere in the normed space of signed measures and the finite measures.

Fan triangulation of a convex polygon Fan triangulation of a concave polygon with a unique concave vertex. A fan triangulation is a simple way to triangulate a polygon by choosing a vertex and drawing diagonals to all of the other vertices of the polygon. Not every polygon can be triangulated this way, so this method is usually only used for convex polygons.

The length of the shell varies from 12 mm to 20 mm. The ovate, conical shell is pointed at the summit. The pyramidal spire is formed of six or seven distinct, smooth, convex whorls. These are covered with very prominent, convex, longitudinal folds, intersected only at the base, and upon the two or three upper whorls, by a few pretty deep transverse striae.

Stipe and cap of the mushroom The cap is white and dry, measuring wide, and convex in shape (conico- or plano-convex). It often has a broad low umbo. The cap's flesh may be thick. At first the cap is covered by the soft, white fragmentary remains of the universal veil, which become more widely separated as the cap expands.

The length of the shell varies from 15 mm to 30 mm. The ovate, conical shell is pointed at the summit . The pyramidal spire is formed of six or seven distinct, smooth, convex whorls. These are covered with very prominent, convex, longitudinal folds, intersected only at the base, and upon the two or three upper whorls, by a few pretty deep transverse striae.

The length of the shell attains 14.4 mm, its width 5.2 mm. (Original description) The solid shell is white, high, narrow, conical, with a blunt apex and a rounded base. It contains 9½ whorls, including a protoconch of 2½ convex whorls, the first two smooth, the rest faintly subdistantly axially plicate, ending abruptly. The whorls of the spire are convex.

Ornithosuchids are unique among crurotarsans, and all other archosaurs, in their possession of a "crocodile-reversed" ankle. In a "crocodile-reversed" ankle, the placement of the concavity is reversed: instead of being on the calcaneum, it is on the astragalus. In ornithosuchids, the calcaneum bears a convex projection that is analogous to the convex projection on the "crocodile-normal" astragalus.

Minkowski's geometry of numbers had a profound influence on functional analysis. Minkowski proved that symmetric convex bodies induce norms in finite-dimensional vector spaces. Minkowski's theorem was generalized to topological vector spaces by Kolmogorov, whose theorem states that the symmetric convex sets that are closed and bounded generate the topology of a Banach space.For Kolmogorov's normability theorem, see Walter Rudin's Functional Analysis.

The method has links to the method of multipliers and dual ascent method and multiple generalizations exist. One drawback of the method is that it is only provably convergent if the objective function is strictly convex. In case this can not be ensured, as for linear programs or non-strictly convex quadratic programs, additional methods such as proximal gradient methods have been developed.

The simplest rectilinear polygon is an axis- aligned rectangle - a rectangle with 2 sides parallel to the x axis and 2 sides parallel to the y axis. See also: Minimum bounding rectangle. A golygon is a rectilinear polygon whose side lengths in sequence are consecutive integers. A rectilinear polygon which is not a rectangle is never convex, but it can be orthogonally convex.

Astyanax pelecus has a compressed and elongate body; the greatest body depth is located anterior to its dorsal fin's origin. The tip of the supraoccipital spine is straight or slightly convex. The profile of its body is convex from the tip of the aforementioned spine to the base of the last dorsalfin ray. The profile along the anal fin's base is posterodorsally slanted.

Astyanax microschemos has a compressed and elongate body; the greatest body depth is located anterior to its dorsal fin's origin. The tip of the supraoccipital spine is straight or slightly convex. The profile of its body is convex from the tip of the aforementioned spine to the base of the last dorsalfin ray. The profile along the anal fin's base is posterodorsally slanted.

The Gestalt psychologists demonstrated that we tend to perceive as figures those parts of our perceptual fields that are convex, symmetric, small, and enclosed.

Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature. Invent. Math. 84 (1986), no. 3, 463–480.Richard Schoen and Shing Tung Yau.

In short, it states the bounded subsets are the same for any two Hausdorff locally convex topologies that are compatible with the same duality.

The species are 8–20 mm. in length. Ratio of body length to greatest body width 2.5–4.55. Body slightly flattened to moderately convex.

The convex leaf shape may be useful for interior leaves which depend on capturing reflected light scattered in random directions from the outer canopy.

In 1930, Funk introduced a non-symmetric metric. It is defined in a domain bounded by a closed convex hypersurface and is also flat.

The extreme points of a compact convex form a Baire space (with the subspace topology) but this set may fail to be closed in .

An inductive limit in the category of locally convex TVSs of a family of bornological (resp. barrelled, quasi-barrelled) spaces has this same property.

The apex is acute. The sutures are impressed. The six whorls are convex, encircled by numerous, close fine striae. The periphery is obtusely angular.

The minute apex is acute. The sutures impressed. There are about 8 or 9, convex whorls. The body whorl is rounded at the periphery.

The 2½ whorls are convex. The suture is very distinct though not deep. The spire is a little raised. The aperture is nearly circular.

However, for non-convex polyhedra, there may not exist a plane near the vertex that cuts all of the faces incident to the vertex.

Solem collaborated on the development of pseudo characteristic functions of convex polyhedra, a result providing rapid regional particle location in Monte Carlo calculations (2003b).

The base of the shell is convex. The aperture is circular. The columella is subdentate at its base. The thick columellar calluses whitish-green.

The shell has an orbiculate-conoid shape. It is dirty red with white spots. The transverse ribs are granulated. The whorls are slightly convex.

The surface is microscopically spirally densely striate. The slender spire is straight-sided. The apex is acute. The 7 whorls are a little convex.

It contains oxbow lakes from former meanders. The population practices agriculture on the convex margins of the river, with their homes slightly higher up.

The width of the shell is 2.5 mm.; the height of the shell is 4.6 mm. Spire short, and flat sided. Whorls weakly convex.

In integral geometry (otherwise called geometric probability theory), Hadwiger's theorem characterises the valuations on convex bodies in Rn. It was proved by Hugo Hadwiger.

The head has a convex area with combs and spines on it. The thorax of these species of Cleopsylla have different rows of setae.

Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens. Michigan Math. J. 5 (1958), 105–126.Moser, Jürgen.

Show slight sexual dimorphism. Male is about 7.9mm in length. Body elongate-oval and convex. A glossy black beetle with reddish to greenish sheen.

One can maximize any quasiconvex combination of weights by finding and checking each convex hull vertex, often more efficiently than checking all possible solutions.

The material is praised for its colors, its pleasing sound as compared to glass or to synthetics such as melamine, and its lower cost as opposed to other materials such as slate/shell. The term "yunzi" can also refer to a single-convex stone made of any material; however, most English- language Go suppliers specify Yunzi as a material and single-convex as a shape to avoid confusion, as stones made of Yunzi are also available in double- convex while synthetic stones can be either shape. Traditional stones are made so that black stones are slightly larger in diameter than white; this is to compensate for the optical illusion created by contrasting colors that would make equal-sized white stones appear larger on the board than black stones. An example of single-convex stones and Go Seigen bowls.

However, the upper surfaces are not bleached, and the outer margins of the forewings of the male specimens are more convex than in Lepidochrysops praeterita.

The spire is low, sides convex. The suture is shallow, adpressed. The shell has 6 whorls, that are rapidly increasing. The last whorl is rounded.

The Brunn-Minkowski inequality gives much insight into the geometry of high dimensional convex bodies. In this section we sketch a few of those insights.

Male frons distinctly concave as opposed to the convex female frons. Asian species can be easily separated from African counterparts by having sexually dimorphic antennae.

If is a Hausdorff locally convex space then the canonical injection from into its bidual is a topological embedding if and only if is infrabarrelled.

A body whorl is subrounded, its base a little convex. It is sculptured with granose cinguli. The aperture is subrotund. The lip is lirate within.

Return boomerangs have a flat convex surface that must be thrown upright with a sharp flick of the wrist, but throwing sticks are thrown horizontally.

Body and head mottled greyish brown. predorsal profile convex uniformly. Two pairs of barbels present. Mouth terminal where lower jaw longer than the upper jaw.

Also, the anal fin has a broadly convex margin rather than a straight margin. Cetopsidium species are smaller than Cetopsis species, growing to only SL.

It represents the exterior envelope of a vertex-centered orthogonal projection of the 600-cell, one of six convex regular 4-polytopes, into 3 dimensions.

The shells of species in this genus have a relatively high cyrtoconoid (approaching a conical shape but with convex sides) spire and a siphonal notch.

A saddle joint (sellar joint, articulation by reciprocal reception) is a type of synovial joint in which the opposing surfaces are reciprocally concave and convex.

These are convex and rounded. The body whorl is equal to about two fifths of the spire. The apex is twisted. The suture is distinct.

Every convex polygon with area T can be inscribed in a triangle of area at most equal to 2T. Equality holds (exclusively) for a parallelogram.

In mathematics, a function f is logarithmically convex or superconvexKingman, J.F.C. 1961. A convexity property of positive matrices. Quart. J. Math. Oxford (2) 12,283-284.

Therefore, every convex combination of points of X belongs to a simplex whose vertices belong to X, and the third and fourth definitions are equivalent.

Corrective lenses provide a range of vision correction, some as high as +4.0 diopter. People with presbyopia choose convex lens for reading glasses; specialized preparations of convex lens usually require the services of an optometrist. Contact lenses can also be used to correct the focusing loss that comes along with presbyopia. Multifocal contact lenses can be used to correct vision for both the near and the far.

The two apical whorls are smooth, convex, rather large. The rest is considerably excavated above and rather bulgingly convex inferiorly and obliquely ribbed. There are 9 ribs on the penultimate whorl, subobsolete in the concavity at the upper part of the whorl, and again nodulous at the suture. The body whorl shows a transverse series of white dots on the ribs a little below the middle.

The geometric notions of an affine space, projective space, convex set, and cone have related notions of basis. An affine basis for an n-dimensional affine space is n+1 points in general linear position. A ' is n+2 points in general position, in a projective space of dimension n. A ' of a polytope is the set of the vertices of its convex hull.

Epinephelus ongus has a body which has a standard length that is 2.7 to 3.2 times as long as it is deep. The dorsal profile of the head is moderately convex, while the area between the eyes is flat. The preopercle is rounded and the serrations on its edge are largely clothed in skin. The upper edge of the gill cover is notably convex.

The number of vertices in a proprism is equal to the product of the number of vertices in all the polytopes in the product. The minimum symmetry order of a proprism is the product of the symmetry orders of all the polytopes. A higher symmetry order is possible if polytopes in the product are identical. A proprism is convex if all its product polytopes are convex.

In mathematics, the Milman–Pettis theorem states that every uniformly convex Banach space is reflexive. The theorem was proved independently by D. Milman (1938) and B. J. Pettis (1939). S. Kakutani gave a different proof in 1939, and John R. Ringrose published a shorter proof in 1959. Mahlon M. Day (1941) gave examples of reflexive Banach spaces which are not isomorphic to any uniformly convex space.

Albert Van Helden, Sven Dupré, Rob van Gent, The Origins of the Telescope, Amsterdam University Press, 2010, page 183 Galileo's telescope used a convex objective lens and a concave eye lens, a design is now called a Galilean telescope. Johannes Kepler proposed an improvement on the designSee his books Astronomiae Pars Optica and Dioptrice that used a convex eyepiece, often called the Keplerian Telescope.

As a special case, the open interval is defined as the cut . More generally, a proper subset S of K is called convex if it contains an interval between every pair of points: for , either or must also be in S. A convex set is linearly ordered by the cut for any not in the set; this ordering is independent of the choice of .

In this species, the exoskeleton was ovate, increasing its width from the prosoma to the fifth segment and rapidly decreasing posteriorly. It had a convex shape, with the second and third segments being the most convex. The prosoma was smooth, short, rounded on the front and with somewhat concave posterior borders. It was 1.7 cm (0.7 in) long and 4.2 cm (1.6 in) wide.

That is, whenever a graph is both planar and 3-vertex-connected, there exists a polyhedron whose vertices and edges form an isomorphic graph.Lectures on Polytopes, by Günter M. Ziegler (1995) , Chapter 4 "Steinitz' Theorem for 3-Polytopes", p.103.. Given such a graph, a representation of it as a subdivision of a convex polygon into smaller convex polygons may be found using the Tutte embedding..

The fourth distal tarsal is enlarged, with its proximal articular surface facing the convex. The convex is articulated by the astragalus- calcaneum complex. This morphology indicates a highly mobile mesotarsal joint in both Varanops and Mycterosaurus, contrasting earlier beliefs that little movement was present in early synapsids. These observations serve as evidence to suggest that Varanops and Mycterosaurus used a semidigitigrade stance to ambulate.

An example of a concave polygon. A simple polygon that is not convex is called concave,. non-convex or reentrant.. A concave polygon will always have at least one reflex interior angle—that is, an angle with a measure that is between 180 degrees and 360 degrees exclusive. Some lines containing interior points of a concave polygon intersect its boundary at more than two points.

The convex polytope therefore is an m-dimensional manifold with boundary, its Euler characteristic is 1, and its fundamental group is trivial. The boundary of the convex polytope is homeomorphic to an (m − 1)-sphere. The boundary's Euler characteristic is 0 for even m and 2 for odd m. The boundary may also be regarded as a tessellation of (m − 1)-dimensional spherical space -- i.e.

The shell contains 7 whorls, angularly convex, finely spirally striated throughout and longitudinally regularly ribbed. The ribs are narrow, rather distant (12 on the penultimate whorl). The body whorl is longer than the spire, angular above, then slightly convex, attenuated towards the base, terminating in a short narrow slightly recurved rostrum. The aperture is long, rather wide in the middle, and narrower at each end.

The base of the shell is very flatly rounded with 7 concentric narrow lirae, the inner 4 closer than the rest, which are separated by 4 to 6 interlirate striae. The umbilicus is narrow, minutely axially incised. The aperture is oblique and roundly quadrate. The outer lip is slightly convex, thin, and smooth within The margin is sinuously convex below the suture, and concave towards the periphery.

The cap is 4.5–16 (18) cm wide, convex, and becomes broadly convex to flat in age. It is bright yellow or yellow-orange, usually more orange or reddish orange towards the disc, and fading to pale yellow. The volva is distributed over the cap as cream to pale tan warts; it is otherwise smooth and sticky when wet. The margin becomes slightly striate in age.

In a vector bornology, is bornivorous if it absorbs every bounded balanced set and in a convex vector bornology is bornivorous if it absorbs every bounded disk. Two TVS topologies on the same vector space have that same bounded subsets if and only if they have the same bornivores. Every bornivorous subset of a locally convex metrizable topological vector space is a neighborhood of the origin.

A fractional program in which f is nonnegative and concave, g is positive and convex, and S is a convex set is called a concave fractional program. If g is affine, f does not have to be restricted in sign. The linear fractional program is a special case of a concave fractional program where all functions f, g, h_j, j=1, \ldots, m are affine.

Not every planar graph corresponds to a convex polyhedron in this way: the trees do not, for example. Steinitz's theorem says that the polyhedral graphs formed from convex polyhedra are precisely the finite 3-connected simple planar graphs. More generally, Euler's formula applies to any polyhedron whose faces are simple polygons that form a surface topologically equivalent to a sphere, regardless of its convexity.

Many observers see Figure 8 as "flipping in and out" between a convex cube and a concave "corner". This corresponds to the two possible orientations of the space. Seeing the figure as convex gives a left-handed coordinate system. Thus the "correct" way to view Figure 8 is to imagine the x-axis as pointing towards the observer and thus seeing a concave corner.

Bone, (45). 799-807. The concave side of a vertebra is less porous and has a thicker cortical bone than the convex side, which is consistent with Wolff's law about bone remodeling.A Comparison of the Microarchitectural Bone Adaptations of the Concave and Convex Thoracic Spinal Facets in Idiopathic Scoliosis; Kevin G. Shea, Tyler Ford, BS, Roy D. Bloebaum, Jacques D’Astous, and Howard King, 2004.

The aperture is white and the lip is bordered with dark brown. Faint spiral striae sculpture the embryonic whorls, and later whorls are convex and irregularly wrinkled in the direction of growth-lines. The whorls are convex and the last is often very obtusely angular at the periphery. The aperture is strongly oblique and the lip thickened within by a strong rib near the margin.

SSIM satisfies the non-negativity, identity of indiscernibles, and symmetry properties, but not the triangle inequality, and thus is not a distance metric. However, under certain conditions, SSIM may be converted to a normalized root MSE measure, which is a distance metric. The square of such a metric is not convex, but is locally convex and quasiconvex, making SSIM a feasible target for optimization.

The Mackey–Arens theorem, named after George Mackey and Richard Arens, characterizes all possible dual topologies on a locally convex space. The theorem shows that the coarsest dual topology is the weak topology, the topology of uniform convergence on all finite subsets of X', and the finest topology is the Mackey topology, the topology of uniform convergence on all absolutely convex weakly compact subsets of X'.

Let E be a locally convex topological vector space. Let C be a compact convex subset of E. Let S be a commuting family of self-mappings T of C which are continuous and affine, i.e. T(tx +(1 – t)y) = tT(x) + (1 – t)T(y) for t in [0,1] and x, y in C. Then the mappings have a common fixed point in C.

In the above form, the functional to be extended must already be bounded by a sublinear function. In some applications, this might close to begging the question. However, in locally convex spaces, any continuous functional is already bounded by the norm, which is sublinear. One thus hasIn category-theoretic terms, the field is an injective object in the category of locally convex vector spaces.

In quantum physics, the state space of any quantum system — the set of all ways the system can be prepared — is a convex hull whose extreme points are positive-semidefinite operators known as pure states and whose interior points are called mixed states. The Schrödinger–HJW theorem proves that any mixed state can in fact be written as a convex combination of pure states in multiple ways.

The space of oriented lines is a double cover of the space of unoriented lines. The Crofton formula is often stated in terms of the corresponding density in the latter space, in which the numerical factor is not 1/4 but 1/2. Since a convex curve intersects almost every line either twice or not at all, the unoriented Crofton formula for convex curves can be stated without numerical factors: the measure of the set of straight lines which intersect a convex curve is equal to its length. The Crofton formula generalizes to any Riemannian surface; the integral is then performed with the natural measure on the space of geodesics.

The veins of insect wings are characterized by a convex-concave placement, such as those seen in mayflies (i.e., concave is "down" and convex is "up") which alternate regularly and by its triadic type of branching; whenever a vein forks there is always an interpolated vein of the opposite position between the two branches. A concave vein will fork into two concave veins (with the interpolated vein being convex) and the regular alteration of the veins is preserved. The veins of the wing appear to fall into an undulating pattern according to whether they have a tendency to fold up or down when the wing is relaxed.

He was appointed Professor of Mathematical Models in Physics at Poitiers University and later Professor of General Mechanics at Monpellier University II. He was emeritus professor in the Laboratoire de Mécanique et Génie Civil, a joint research unit of the university and the CNRS. Moreau's principal works have been in non-smooth mechanics and convex analysis. He is considered one of the founders of convex analysis, where several fundamental and now classical results have his name (Moreau's lemma of the two cones, Moreau's envelopes, Moreau-Yosida's approximations, Fenchel-Moreau's theorem, etc.). He founded the Convex Analysis Group in the 1970s at Montpellier University (France).

By definition, it is the coarsest topology on for which all maps in are continuous. That the vector space operations are continuous in this topology follows from properties 2 and 3 above. It can easily be seen that the resulting topological vector space is "locally convex" in the sense of the first definition given above because each is absolutely convex and absorbent (and because the latter properties are preserved by translations). Note that it is possible for a locally convex topology on a space to be induced by a family of norms but for to not be normable (that is, to have its topology be induced by a single norm).

The elongated square gyrobicupola (J37), a Johnson solid 24 equilateral triangle example is not a Johnson solid because it is not convex. This 24-square example is not a Johnson solid because it is not strictly convex (has 180° dihedral angles.) In geometry, a Johnson solid is a strictly convex polyhedron such that each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides (J1); it has 1 square face and 4 triangular faces.

A convex polytope can be decomposed into a simplicial complex, or union of simplices, satisfying certain properties. Given a convex r-dimensional polytope P, a subset of its vertices containing (r+1) affinely independent points defines an r-simplex. It is possible to form a collection of subsets such that the union of the corresponding simplices is equal to P, and the intersection of any two simplices is either empty or a lower-dimensional simplex. This simplicial decomposition is the basis of many methods for computing the volume of a convex polytope, since the volume of a simplex is easily given by a formula.

A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection of edges, faces or facets of the object. In a polygon, a vertex is called "convex" if the internal angle of the polygon (i.e., the angle formed by the two edges at the vertex with the polygon inside the angle) is less than π radians (180°, two right angles); otherwise, it is called "concave" or "reflex". More generally, a vertex of a polyhedron or polytope is convex, if the intersection of the polyhedron or polytope with a sufficiently small sphere centered at the vertex is convex, and is concave otherwise.

The masked grouper has an oblong, rather compressed body in which the standard length is 2.6 to 3.3 times its depth. The dorsal profile of the head is convex while the area between the eyes is slightly convex. The preopercle is rounded with fine serratations and with a smooth, fleshy lower margin. The gill cover has a central spine which is located at one-third of the gap between the lower to upper spines and with an upper edge which is distinctly convex The dorsal fin contains 9 spines and 14-16 soft rays while the anal fin has 3 spines and 9-10 soft rays.

Steiner's proof was completed later by several other mathematicians. Steiner begins with some geometric constructions which are easily understood; for example, it can be shown that any closed curve enclosing a region that is not fully convex can be modified to enclose more area, by "flipping" the concave areas so that they become convex. It can further be shown that any closed curve which is not fully symmetrical can be "tilted" so that it encloses more area. The one shape that is perfectly convex and symmetrical is the circle, although this, in itself, does not represent a rigorous proof of the isoperimetric theorem (see external links).

A Tverberg partition of the vertices of a regular heptagon into three subsets with intersecting convex hulls. In discrete geometry, Tverberg's theorem, first stated by , is the result that sufficiently many points in d-dimensional Euclidean space can be partitioned into subsets with intersecting convex hulls. Specifically, for any set of :(d + 1)(r - 1) + 1\ points there exists a point x (not necessarily one of the given points) and a partition of the given points into r subsets, such that x belongs to the convex hull of all of the subsets. The partition resulting from this theorem is known as a Tverberg partition.

The head of Spectacular Optical, Barry Convex, has been secretly working with Harlan to get Max exposed to Videodrome and to have him broadcast it, as part of a conspiracy to end North America's cultural decay by giving fatal brain tumours to anyone obsessed enough with sex and violence who would watch Videodrome. Convex then inserts a brainwashing Betamax tape into Max's torso. Under Convex's influence, Max murders his colleagues at CIVIC-TV, and later attempts to murder Bianca, but she manages to stop Max by showing him a videotape of Nicki being strangled to death. Bianca then 'reprograms' Max to kill Harlan and Convex on her orders.

In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory. Intuitively, subharmonic functions are related to convex functions of one variable as follows. If the graph of a convex function and a line intersect at two points, then the graph of the convex function is below the line between those points. In the same way, if the values of a subharmonic function are no larger than the values of a harmonic function on the boundary of a ball, then the values of the subharmonic function are no larger than the values of the harmonic function also inside the ball.

Lev M. Bregman (born 31 January 1941 in Leningrad) is a Soviet and Israeli mathematician, most known for the Bregman divergence named after him. Bregman received his M. Sc. in mathematics in 1963 at Leningrad University and his Ph.D. in mathematics in 1966 at the same institution, under the direction of his advisor Prof. J. V. Romanovsky, for his thesis about relaxation methods for finding a common point of convex sets, which led to one of his most well- known publications.Brègman, L. M. A relaxation method of finding a common point of convex sets and its application to the solution of problems in convex programming.

The length of mandible is 8.0–9.1 mm and their forearm is less than 41 mm in length. The margin above nostril lobulated and slightly convex.

Their results can also be extended to settings other than the mean curvature flow.Ben Andrews. Evolving convex curves. Calc. Var. Partial Differential Equations 7 (1998), no.

The open siphonal canal is very short. The outer lip is rounded, very convex, incrassate inwards and ending in a V-shape sinus at its top.

If is a Hausdorff locally convex space then the following are equivalent: 1. is reflexive; 2. is semireflexive and infrabarreled; 3. is semireflexive and barreled; 4.

It is easy to prove that an intersection of any collection of orthoconvex sets is orthoconvex. Some other properties of convex sets are valid as well.

The procedure allows the production of parts with relatively exact outer contours. This forming process is suitable for the manufacturing of parts with smooth, convex surfaces.

Abdominal scent pouches are oval and broad with yellowish scales. Forewings narrow and elongate. Costa faintly convex. Pterostigma strong, thickened, and strongly projecting along costal edge.

54, no. 3, pp. 203–220, 2006. The primal-dual approach can also be used to find local optima in cases when f is non-convex.

Philip Starr "Phil" Wolfe (August 11, 1927 – December 29, 2016) was an American mathematician and one of the founders of convex optimization theory and mathematical programming.

The shell contains four whorls. The first two are smooth and very rapidly increasing. The remainder are convex. They are ornamented with spiral ridges or cords.

The apex is rosy. The spire is short and contains about 5 convex whorls. The rounded-quadrate aperture is iridescent within. The lip is white-margined.

In functional analysis, a locally convex topological vector space is a bornological space if its topology can be recovered from its bornology in a natural way.

In model theory, a weakly o-minimal structure is a model theoretic structure whose definable sets in the domain are just finite unions of convex sets.

Transactions of the American Mathematical Society vol. 353 (2001), no. 3, pp. 943–962.S. Cleary and J. Taback, Thompson's group F is not almost convex.

Every ordered locally convex space is regularly ordered. The canonical orderings of subspaces, products, and direct sums of regularly ordered vector spaces are again regularly ordered.

When P0, …, Pm are all positive- definite matrices, the problem is convex and can be readily solved using interior point methods, as done with semidefinite programming.

For terms see gastropod shell. The shell is yellowish to brownish, smooth or with fine striation. It is shiny. There are 7-8 slightly convex whorls.

It grows to a length of 22.0 cm. Dorsal profile strongly convex. Body brownish with creamy lower third of flanks. Brown spots irregularly distributed on flanks.

The 4½ slightly convex whorls enlarge rapidly. The body whorl is subangulated at the base. The umbilical area is longitudinally crispate. The continuous peristome is thickened.

For each topology 𝜏 on X such that (X, 𝜏) is a locally convex vector lattice, x is a quasi-interior point of its positive cone.

In statistical decision theory, the risk set of a randomized decision rule is the convex hull of the risk points of its underlying deterministic decision rules.

When a convex lens of the right strength is placed in front of the hypermetropic eye, the light rays are refracted to focus on the retina.

In complex analysis, a branch of mathematics, the Gauss–Lucas theorem gives a geometrical relation between the roots of a polynomial P and the roots of its derivative P′. The set of roots of a real or complex polynomial is a set of points in the complex plane. The theorem states that the roots of P′ all lie within the convex hull of the roots of P, that is the smallest convex polygon containing the roots of P. When P has a single root then this convex hull is a single point and when the roots lie on a line then the convex hull is a segment of this line. The Gauss–Lucas theorem, named after Carl Friedrich Gauss and Félix Lucas, is similar in spirit to Rolle's theorem. Illustration of Gauss Lucas theorem, displaying the evolution of the roots of the derivatives of a polynomial.

The pentagram map, when acting on the moduli space X of convex polygons, has an invariant volume form. At the same time, as was already mentioned, the function f=O_NE_N has compact level sets on X. These two properties combine with the Poincaré recurrence theorem to imply that the action of the pentagram map on X is recurrent: The orbit of almost any equivalence class of convex polygon P returns infinitely often to every neighborhood of P. This is to say that, modulo projective transformations, one typically sees nearly the same shape, over and over again, as one iterates the pentagram map. (It is important to remember that one is considering the projective equivalence classes of convex polygons. The fact that the pentagram map visibly shrinks a convex polygon is irrelevant.) It is worth mentioning that the recurrence result is subsumed by the complete integrability results discussed below.

Plano-convex ingots are lumps of metal with a flat or slightly concave top and a convex base. They are sometimes, misleadingly, referred to as bun ingots which imply the opposite orientation of concavity.Weisgerber, G. and Yule, P (2003) ‘Al Aqir near Bahla–an Early Bronze Age Dam Site with Plano Convex Ingots’ in Arabian Archaeology and Epigraphy 14 p.48 They are most often made of copper although plano-convex ingots of other materials such as copper alloy, lead and tin are also known.Bass, G.F. (1967) Cape Gelidonya: A Bronze Age Shipwreck US: American Philosophical Society, Maddin, R. & Merkel, J. (1990) ‘Metallographic and statistical analyses’ in Lo Schiavo, F Maddin, R Merkel, J. Muhly, J. D. & Stech, T (eds) Metallographic and statistical analyses of copper ingots from Sardinia P.42-199. Ozieri: Il Torchietto Tylecote, R.F. (1987) The Early History of Metallurgy in Europe.

One structure that can be used for this purpose is the convex layers of the input point set, a family of nested convex polygons consisting of the convex hull of the point set and the recursively-constructed convex layers of the remaining points. Within a single layer, the points inside the query half-plane may be found by performing a binary search for the half-plane boundary line's slope among the sorted sequence of convex polygon edge slopes, leading to the polygon vertex that is inside the query half-plane and farthest from its boundary, and then sequentially searching along the polygon edges to find all other vertices inside the query half-plane. The whole half-plane range reporting problem may be solved by repeating this search procedure starting from the outermost layer and continuing inwards until reaching a layer that is disjoint from the query halfspace. Fractional cascading speeds up the successive binary searches among the sequences of polygon edge slopes in each layer, leading to a data structure for this problem with space O(n) and query time O(log n + h).

In a convex quadrilateral, the quasiorthocenter H, the "area centroid" G, and the quasicircumcenter O are collinear in this order on the Euler line, and HG = 2GO..

The whorls are very slightly convex and angulated at the sutures. The aperture is very oval. The colour of the shell is very white.Pease, W. H. (1860).

The base is moderately convex. The columella is short, smooth and obliquely truncate. The outer lip is thin and sharp. The siphonal canal is wide and short.

The size of the shell varies between 27 mm and 78 mm. The shell is obsolete!y coronated with tubercles. The stout body whorl is somewhat convex.

During convex plasmolysis, the plasma membrane and the enclosed protoplast shrinks completely from the cell wall, with the plasma membrane's ends in a symmetrically, spherically curved pattern.

Wolfe pioneered bundle methods of descent for convex minimization.Citation of Claude Lemaréchal for the George Dantzig Prize in 1994 in Optima, Issue 44 (1994) pages 4-5.

The proof uses a fixed-point theorem by Eilenberg and Montgomery. Note: Every convex set is contractible, so Diamantaras' theorem is more general than the previous three.

The Exact Feasibility of Randomized Solutions of Uncertain Convex Programs. SIAM J. on Optimization, 19, no.3: 1211-1230, 2008.A. Caré, S. Garatti and M.C. Campi.

Self-portrait in a Convex Mirror (c. 1524) is a painting by the Italian late Renaissance artist Parmigianino. It is housed in the Kunsthistorisches Museum, Vienna, Austria.

The earlier 'Tribute Edition', a limited run of 1200 units, featured blue LED lighting, wooden side panels and Bob Moog's signature decaled onto the convex back panel.

The cardinal and ordinal definitions are equivalent in the case of a convex consumption set with continuous preferences that are locally non-satiated in the first argument.

The sutures are impressed. The 6 whorls are convex, the last obtusely angled, flattened beneath. The aperture is very oblique, rounded, iridescent inside. The peristome is simple.

The male has a convex carapace long and an abdomen long. It is generally brown in colour, with white patterns on the abdomen and some yellow legs.

Shell of Pseudothurmannia species can reach a diameter of about . They show flat or slightly convex sides, a surface with dense ribs and a subquadrate whorl section.

The peristome is simple. The white columella is thick and smooth. The operculum is almost round. Its outer surface is white, very convex and everywhere minutely granulate.

The prototypical primal and dual semidefinite program is a minimization of a real linear function respectively subject to the primal and dual convex cones governing this LMI.

The simplest form of treatment for far-sightedness is the use of corrective lenses, eyeglasses or contact lenses. Eyeglasses used to correct far- sightedness have convex lenses.

The eight whorls of the teleoconch are convex. They contain a deep suture. They are longitudinally and spirally ribbed. The interstices of the decussations appear as pitted.

The sutures are hardly impressed. The body whorl is slightly convex with 7 - 8 carinae. The small aperture is elongate-oval. The outer lip is deeply sinuated.

A proof runs as follows: let be the convex hull of }. Note that is an absorbing disk in , and call its Minkowski functional . Then on and on .

Very similar to Liang–Barsky line-clipping algorithm. The difference is that Liang–Barsky is a simplified Cyrus–Beck variation that was optimized for a rectangular clip window. The Cyrus–Beck algorithm is primarily intended for a clipping a line in the parametric form against a convex polygon in 2 dimensions or against a convex polyhedron in 3 dimensions.Cyrus, M., Beck, J.: Generalized Two and Three Dimensional Clipping, Computers & Graphics, Vol.

Melencolia I by Albrecht Dürer, the first appearance of Dürer's solid (1514). In the mathematical field of graph theory, the Dürer graph is an undirected graph with 12 vertices and 18 edges. It is named after Albrecht Dürer, whose 1514 engraving Melencolia I includes a depiction of Dürer's solid, a convex polyhedron having the Dürer graph as its skeleton. Dürer's solid is one of only four well-covered simple convex polyhedra.

The holotype skeleton shows each structure attaching to a vertebral spine. These anchorage points are visible as raised knobs. The base of each appendage is slightly convex, unlike the flattened shape of the rest of the structure. The convex shape may be evidence that the base of each structure was tubular in life, anchoring like other integumentary structures such as mammalian hair or avian feathers into a follicle.

The largest strictly-convex deltahedron is the regular icosahedron This is a truncated tetrahedron with hexagons subdivided into triangles. This figure is not a strictly-convex deltahedron since coplanar faces are not allowed within the definition. In geometry, a deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek upper case delta (Δ), which has the shape of an equilateral triangle.

The length of the shell attains 18.6 mm, its diameter 4.5 mm. (Original description) The solid, narrow shell has an elongate-fusiform shape. It consists of 9 whorls, including the protoconch of 3 convex smooth whorls, with a deep impressed suture. The -whorls of the spire are convex, roundly angled below the middle in the early whorls, above it in the later, slightly adpressed below the linear suture.

There is no single straightforward generalization of polygon monotonicity to higher dimensions. In one approach the preserved monotonicity trait is the line L. A three-dimensional polyhedron is called weakly monotonic in direction L if all cross-sections orthogonal to L are simple polygons. If the cross-sections are convex, then the polyhedron is called weakly monotonic in convex sense. Both types may be recognized in polynomial time.

The length of the shell attains 8¼ mm, its diameter 3 mm. (Original description) The white shell is elongately-fusiform, thin and transparent. It contains about 9 whorls, of which about 3 form a reddish-brown protoconch, composed of convex whorls, with riblets in different directions, but the protoconch being rather worn, the sculpture is not prominent. The subsequent whorls are convex, with a narrow, excavated part below the deep suture.

The pointed spire is conical and, formed of six slightly convex whorls, the lowest of which is as large as all the others. They are flattened and angular at the upper part, crowned upon the angle by a subgranulated margin. The suture is accompanied at the upper part of each whorl, by a small, slightly convex and undulating margin. Upon the body whorl are seen nine rounded, transverse, very angular folds.

This means that the affine combination belongs to , for all and in , and in the interval . This implies that convexity (the property of being convex) is invariant under affine transformations. This implies also that a convex set in a real or complex topological vector space is path-connected, thus connected. A set is ' if every point on the line segment connecting and other than the endpoints is inside the interior of .

For example, a Voronoi diagram is commonly represented by a DCEL inside a bounding box. This data structure was originally suggested by Muller and PreparataMuller, D. E.; Preparata, F. P. "Finding the Intersection of Two Convex Polyhedra", Technical Report UIUC, 1977, 38pp, also Theoretical Computer Science, Vol. 7, 1978, 217–236 for representations of 3D convex polyhedra. Later, a somewhat different data structure was suggested, but the name "DCEL" was retained.

The 5 or 6 subcircular perforations are open with their edges moderately prominent..The right margin is quite convex, especially in the part of the lip adjacent to the spire. The back of the shell is convex. It is not carinated at the row of holes, but there is a shallow sulcus just below it. The color of the shell is a reddish-brown, with irregular zigzagly radiating white flames.

Plasmolysis is mainly known as shrinking of cell membrane in hypertonic solution and great pressure. Plasmolysis can be of two types, either concave plasmolysis or convex plasmolysis. Convex plasmolysis is always irreversible while concave plasmolysis is usually reversible. During concave plasmolysis, the plasma membrane and the enclosed protoplast partially shrinks from the cell wall due to half-spherical, inwarding curving pockets forming between the plasma membrane and the cell wall.

He also worked in numerical analysis and on ill-posed problems. His textbook on partial differential equations was highly influential and was re-edited many times. He made several contributions to convex geometry, including his famous result that within every convex body there is one ellipsoid of maximal volume, now called the John Ellipsoid. From the mid-1950s on, he started working on the theory of equilibrium nonlinear elasticity.

1\. If there is only a single commodity type, then any weakly-monotonically- increasing preference relation is convex. This is because, if y \geq x , then every weighted average of y and ס is also \geq x . 2\. Consider an economy with two commodity types, 1 and 2. Consider a preference relation represented by the following Leontief utility function: :u(x_1,x_2) = \min(x_1,x_2) This preference relation is convex.

The iterates of the pentagram map shrink any convex polygon exponentially fast to a point. This is to say that the diameter of the nth iterate of a convex polygon is less than K a^n for constants K>0 and 0 which depend on the initial polygon. Here we are taking about the geometric action on the polygons themselves, not on the moduli space of projective equivalence classes of polygons.

The lower bound on worst- case running time of output-sensitive convex hull algorithms was established to be Ω(n log h) in the planar case. There are several algorithms which attain this optimal time complexity. The earliest one was introduced by Kirkpatrick and Seidel in 1986 (who called it "the ultimate convex hull algorithm"). A much simpler algorithm was developed by Chan in 1996, and is called Chan's algorithm.

Helly's theorem for the Euclidean plane: if a family of convex sets has a nonempty intersection for every triple of sets, then the whole family has a nonempty intersection. Helly's theorem is a basic result in discrete geometry on the intersection of convex sets. It was discovered by Eduard Helly in 1913,. but not published by him until 1923, by which time alternative proofs by and had already appeared.

And its lateral margin of the body is gently convex anteroposteriorly while the medial margin is more strongly convex. The coracoid bone of Pistosaurus is flat and expanded medially. The glenoid region is similar to Nothosaurus in development: both the slight notching of its margin and a distinct facet contact with the humeral head. There is also a ridge like thickening which links the glenoid to posteromedial region of the coracoid.

Note that P need not be itself convex. A consequence of this is that P′ can always be extremal in P, as non-extremal points can be removed from P without changing the membership of x in the convex hull. The similar theorems of Helly and Radon are closely related to Carathéodory's theorem: the latter theorem can be used to prove the former theorems and vice versa. See in particular p.

The dorsal margin from the umbo to the upper end of the anterior margin is slightly convex, and from the umbo to the posterior region is straight and sloping. The lower margin is convex and evenly curved, and the surface of the upper part of the shell is striated with fine, irregular, transverse concentric markings. The right valve measures in breadth and in height, and the right valve and .

Even though every equilibrium is efficient, neither of the above two theorems say anything about the equilibrium existing in the first place. To guarantee that an equilibrium exists, it suffices that consumer preferences be strictly convex. With enough consumers, the convexity assumption can be relaxed both for existence and the second welfare theorem. Similarly, but less plausibly, convex feasible production sets suffice for existence; convexity excludes economies of scale.

Every (bounded) convex polytope is the image of a simplex, as every point is a convex combination of the (finitely many) vertices. However, polytopes are not in general isomorphic to simplices. This is in contrast to the case of vector spaces and linear combinations, every finite-dimensional vector space being not only an image of, but in fact isomorphic to, Euclidean space of some dimension (or analog over other fields).

The fruit bodies have convex to broadly convex caps, sometimes with a broad umbo, and measure in diameter. Its color is pale gray, sometimes with tints of brown in the center. The dry cap surface is made of densely interwoven fibrils in the center, and radiating interwoven fibrils elsewhere; there are scattered squamules. The flesh is pale gray, with no distinctive odor and a hot, peppery or bitter taste.

They are rather convex and slightly angular by the projection of one of the numerous lirae. These number about 6 on the penultimate whorl, and 7 on the upper part of the body whorl, besides several intermediate, elevated striae, which become very numerous towards the aperture. The body whorl is angular at the periphery. Its convex base has numerous, crowded, elevated striae (partly wanting in a few specimens).

For -1\leq p \leq 0, the function f(t) = -t^p is operator monotone and operator concave. For 0 \leq p \leq 1, the function f(t) = t^p is operator monotone and operator concave. For 1 \leq p \leq 2, the function f(t) = t^p is operator convex. Furthermore, :f(t) = \log(t) is operator concave and operator monotone, while :f(t) = t \log(t) is operator convex.

Epinephelus heniochus has a body which has a standard length that is 2.7 to 3.1 times its depth. The area between the eyes is slightly convex whereas the dorsal profile of the head is markedly convex. The preopercle is sharply angled with 2-4 large spines located at its angle. The dorsal fin contains 11 spines and 14-15 sot rays while the anal fin has 3 spines and soft rays.

The cap is 4–13 cm wide, hemispheric to convex when young, becoming plano-convex to plano-depressed in age. It is pinkish-melon-colored to peach-orange, sometimes pastel red towards the disc. The cap is slightly appendiculate. The volva is distributed over the cap as thin pale yellowish to pale tannish warts; it is otherwise smooth and subviscid, and the margin becomes slightly to moderately striate in age.

It is also possible to define related flip graphs for partitions into quadrilaterals or pseudotriangles, and for higher-dimensional triangulations. The flip graph of triangulations of a convex polygon forms the skeleton of the associahedron or Stasheff polytope. The flip graph of the regular triangulations of a point set (projections of higher-dimensional convex hulls) can also be represented as a skeleton, of the so-called secondary polytope.

A convex quadrilateral is equidiagonal if and only if its Varignon parallelogram, the parallelogram formed by the midpoints of its sides, is a rhombus. An equivalent condition is that the bimedians of the quadrilateral (the diagonals of the Varignon parallelogram) are perpendicular. A convex quadrilateral with diagonal lengths p and q and bimedian lengths m and n is equidiagonal if and only if :pq=m^2+n^2.

Convex hull, alpha shape and minimal spanning tree of a bivariate data set. In computational geometry, an alpha shape, or α-shape, is a family of piecewise linear simple curves in the Euclidean plane associated with the shape of a finite set of points. They were first defined by . The alpha-shape associated with a set of points is a generalization of the concept of the convex hull, i.e.

The stipe is covered with irregularly shaped glandular dots. The cap of S. pungens is roughly convex when young, becoming plano-convex (flat on one side and rounded on the other) to somewhat flat with age, and reaches diameters of . The cap surface is sticky to slimy when moist, becoming shiny when dried. The surface is smooth but is sometimes streaked with the sticky glue-like cap slime when older.

The worker of Paratrechina zanjensis can easily be separated from Paratrechina longicornis based on the presence of erect macrosetae on the scapes. There are several other notable differences between the two species. The propodeal dorsal face of Paratrechina zanjensis is more convex than is observed in Paratrechina longicornis. Similarly, the pronotum and to a lesser degree the mesonotum are more convex in Paratrechina zanjensis, being almost flat in Paratrechina longicornis.

A rendered image showing a dome magnifier over a page of text. A dome magnifier is a dome-shaped magnifying device made of glass or acrylic plastic, used to enlarge words on a page or computer screen. They are plano-convex lenses: the flat (planar) surface is placed on the object to be magnified, and the convex (dome) surface provides the enlargement. They usually provide between 1.8× and 6× magnification.

This corollary is also some called "the Krein-Milman theorem". A particular case of this theorem, which can be easily visualized, states that given a convex polygon, one only needs the corners of the polygon to recover the polygon shape. The statement of the theorem is false if the polygon is not convex, as then there can be many ways of drawing a polygon having given points as corners.

In another version, if both disjoint convex sets are open, then there is a hyperplane in between them, but not necessarily any gap. An axis which is orthogonal to a separating hyperplane is a separating axis, because the orthogonal projections of the convex bodies onto the axis are disjoint. The hyperplane separation theorem is due to Hermann Minkowski. The Hahn–Banach separation theorem generalizes the result to topological vector spaces.

The 13 convex shapes matched with tangram set Over 6500 different tangram problems have been created from 19th century texts alone, and the current number is ever-growing. Fu Traing Wang and Chuan-Chin Hsiung proved in 1942 that there are only thirteen convex tangram configurations (config segment drawn between any two points on the configuration's edge always pass through the configuration's interior, i.e., configurations with no recesses in the outline).

The algorithm of is based on Coxeter's proof using ordered geometry. It performs the following steps: #Choose a point p_0 that is a vertex of the convex hull of the given points. #Construct a line \ell_0 that passes through p_0 and otherwise stays outside of the convex hull. #Sort the other given points by the angle they make with p_0, grouping together points that form the same angle.

An iteration of the ellipsoid method In mathematical optimization, the ellipsoid method is an iterative method for minimizing convex functions. When specialized to solving feasible linear optimization problems with rational data, the ellipsoid method is an algorithm which finds an optimal solution in a finite number of steps. The ellipsoid method generates a sequence of ellipsoids whose volume uniformly decreases at every step, thus enclosing a minimizer of a convex function.

The John ellipsoid is named after the German-American mathematician Fritz John, who proved in 1948 that each convex body in Rn contains a unique circumscribed ellipsoid of minimal volume and that the dilation of this ellipsoid by factor 1/n is contained inside the convex body.John, Fritz. "Extremum problems with inequalities as subsidiary conditions". Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, 187—204.

In convex geometry, Gordan's lemma states that the semigroup of integral points in the dual cone of a rational convex polyhedral cone is finitely generated. In algebraic geometry, the prime spectrum of the semigroup algebra of such a semigroup is, by definition, an affine toric variety; thus, the lemma says an affine toric variety is indeed an algebraic variety. The lemma is named after the German mathematician Paul Gordan (1837–1912).

HEAD - A bit long under the eyes, usually more right than on the convex bevel, wide forehead and slightly convex. CORNER - Medium size, elliptical section, white, flushed with the tips, going out almost horizontally to the sides, turning from a little behind, then turned forward and with the tips up and turned inside out at the last third of its length. BROW - Little salient. SCREWDRIVER PALBEBRAL - Slightly oblique.

M-estimation has been one of the most popular paradigms for robust estimation in robotics and computer vision. Because robust objective functions are typically non-convex (e.g., the truncated least squares loss v.s. the least squares loss), algorithms for solving the non- convex M-estimation are typically based on local optimization, where first an initial guess is provided, following by iterative refinements of the transformation to keep decreasing the objective function.

A kinetic data structure is a data structure used to track an attribute of a geometric system that is moving continuously. For example, a kinetic convex hull data structure maintains the convex hull of a group of n moving points. The development of kinetic data structures was motivated by computational geometry problems involving physical objects in continuous motion, such as collision or visibility detection in robotics, animation or computer graphics.

Bregman's method is an iterative algorithm to solve certain convex optimization problems. The algorithm is a row-action method accessing constraint functions one by one and the method is particularly suited for large optimization problems where constraints can be efficiently enumerated. The original version is due to Lev M. Bregman.Bregman L. "A Relaxation Method of Finding a Common Point of Convex Sets and its Application to Problems of Optimization". Dokl. Akad.

A prototypical flip graph is that of a convex n-gon \pi. The vertices of this graph are the triangulations of \pi, and two triangulations are adjacent in it whenever they differ by a single interior edge. In this case, the flip operation consists in exchanging the diagonals of a convex quadrilateral. These diagonals are the interior edges by which two triangulations adjacent in the flip graph differ.

If P is a (nonconstant) polynomial with complex coefficients, all zeros of P′ belong to the convex hull of the set of zeros of P.Marden (1966), Theorem (6,1).

The pygidium is convex. Its axis is parallel-sided, and does not reach border furrow. Three pairs of pleural furrows may be discernible. The pygidial border is narrow.

The subangular body whorl is depressed above. The base of the shell is convex, with about 8 concentric lirae. Tnere is no umbilical perforation. The aperture is rhomboidal.

The shell shows many opisthocline ribs. The body whorl is at the top slightly concave, but otherwise slightly convex. The columella is delicately contorted. The aperture is elongated.

The spire is sculptured with transverse impressed lines. The body whorl is angulated. The base of the shell is a little convex. The subquadrate aperture is white inside.

The surface is smooth and polished. The aperture is subrhombic with a highly sinuous outer lip that is convex below. Columella has a week tooth at the top.

Kneeboarding is an aquatic sport where the participant is towed kneeling on a buoyant, convex, and hydrodynamically shaped board at a planing speed, most often behind a motorboat.

The suture is subcanaliculate. The body whorl is obtusely bi-angular at the periphery. The base of the shell is somewhat convex. The aperture is rounded and oblique.

The spire is conic. The 6 to 7 whorls are convex. The apex is usually eroded and orange-colored. The body whorl is flattened around the superior portion.

The whorls are regular and convex. Protoconch is smooth and rounded and the aperture is elongate, with a thickened outer lip. This nocturnal snail predates on small invertebrates.

22 (1987), no. 2, pp. 197–210. a notion that led to substantial further study and generalizations.S. Hermiller and J. Meier, Measuring the tameness of almost convex groups.

Rotkowitz, Michael; and Lall, Sanjay. "A Characterization of Convex Problems in Decentralized Control", IEEE Transactions on Automatic Control, Vol. 51, No. 2, February 2006. Accessed June 1, 2018.

Two of the semicircles are necessarily concave, with arbitrary diameters a and b; the third semicircle is convex, with diameter a+b. Some special points on the arbelos.

1. A bounded sequentially complete disk in a Hausdorff TVS is a Banach disk. 2. A Hausdorff locally convex space that is sequentially complete and bornological is ultrabornological.

The fusiform shell is acuminate. The convex whorls of the spire are cancellated with longitudinal ribs and transverse lirae. The aperture is narrow. The outer lip is varicose.

The thin, fragile, oblong shell is shaped like a Haliotis. Its back is convex. It is all over very delicately striated. It is flesh-colored, spotted with red.

Measurements are only known from the holotype. It has a length of 1 mm. Its overall appearance is ovate, very convex, and nitid. The upperside is pitchy black.

Its characteristics are masculinity, strength and solidity. The Doric capital consists of a cushion-like convex moulding known as an echinus, and a square slab termed an abacus.

The best known data structure for the 2-dimensional kinetic convex hull problem is by Basch, Guibas, and Hershberger. This data structure is responsive, efficient, compact and local.

The columellar lip is a little reflected. There is no operculum or umbilicus. The oval animal is convex above and little spiral. It has a large oblong foot.

The axiality of a given convex shape can be approximated arbitrarily closely in sublinear time, given access to the shape by oracles for finding an extreme point in a given direction and for finding the intersection of the shape with a line.. consider the problem of computing the axiality exactly, for both convex and non-convex polygons. The set of all possible reflection symmetry lines in the plane is (by projective duality) a two-dimensional space, which they partition into cells within which the pattern of crossings of the polygon with its reflection is fixed, causing the axiality to vary smoothly within each cell. They thus reduce the problem to a numerical computation within each cell, which they do not solve explicitly. The partition of the plane into cells has O(n^4) cells in the general case, and O(n^3) cells for convex polygons; it can be constructed in an amount of time which is larger than these bounds by a logarithmic factor.

The problem of finding sets of n points minimizing the number of convex quadrilaterals is equivalent to minimizing the crossing number in a straight-line drawing of a complete graph. The number of quadrilaterals must be proportional to the fourth power of n, but the precise constant is not known. It is straightforward to show that, in higher- dimensional Euclidean spaces, sufficiently large sets of points will have a subset of k points that forms the vertices of a convex polytope, for any k greater than the dimension: this follows immediately from existence of convex k-gons in sufficiently large planar point sets, by projecting the higher- dimensional point set into an arbitrary two-dimensional subspace. However, the number of points necessary to find k points in convex position may be smaller in higher dimensions than it is in the plane, and it is possible to find subsets that are more highly constrained.

The specific Kaltenborn mobilization of posterior- anterior gliding with distraction in grade 3 of the CMC joint; the Convex/Concave Rule was applied in each case. In brief, Kaltenborn described these mechanics in terms of the convex-concave rule; the direction of decreased joint gliding in a hypomobile joint and, thus, the appropriate treatment can be deduced by this rule. With movement of a concave joint partner, the glide occurs in the same direction. The form of the joint surface has been considered to induce its gliding/sliding movement; a female (concave) joint surface glides in the same direction as the bone movement, whereas a male (convex) surface glides in the opposite direction of the bone movement.

Such points > are shrouded in eternal darkness—unless we make our consumer a monopsonist > and let him choose between goods lying on a very convex "budget curve" > (along which he is affecting the price of what he buys). In this monopsony > case, we could still deduce the slope of the man's indifference curve from > the slope of the observed constraint at the equilibrium point. For the epigraph to their seventh chapter, "Markets with non-convex preferences and production" presenting , quote John Milton's description of the (non-convex) Serbonian Bog in Paradise Lost (Book II, lines 592–594): > A gulf profound as that Serbonian Bog > Betwixt Damiata and Mount Casius old, > Where Armies whole have sunk. according to Diewert.

In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the (simple) 3-vertex- connected planar graphs (with at least four vertices). That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs. Branko Grünbaum has called this theorem “the most important and deepest known result on 3-polytopes.” The theorem appears in a 1922 paper of Ernst Steinitz, after whom it is named.

If each of the bonding maps f_{i}^{j} is an embedding of TVSs onto proper vector subspaces and if the system is directed by with its natural ordering, then the resulting limit is called a strict (countable) direct limit. In such a situation we may assume without loss of generality that each is a vector subspace of and that the subspace topology induced on by is identical to the original topology on . In the category of locally convex topological vector spaces, the topology on a strict inductive limit of Fréchet spaces can be described by specifying that an absolutely convex subset is a neighborhood of if and only if is an absolutely convex neighborhood of in for every .

For a strictly convex function, the Legendre transformation can be interpreted as a mapping between the graph of the function and the family of tangents of the graph. (For a function of one variable, the tangents are well-defined at all but at most countably many points, since a convex function is differentiable at all but at most countably many points.) The equation of a line with slope and -intercept is given by . For this line to be tangent to the graph of a function at the point requires :f\left(x_0\right) = p x_0 + b and :p = f'(x_0). The function f'is strictly monotone as the derivative of a strictly convex function.

By forgetting the face structure, any polyhedron gives rise to a graph, called its skeleton, with corresponding vertices and edges. Such figures have a long history: Leonardo da Vinci devised frame models of the regular solids, which he drew for Pacioli's book Divina Proportione, and similar wire-frame polyhedra appear in M.C. Escher's print Stars. Coxeter's analysis of Stars is on pp. 61–62. One highlight of this approach is Steinitz's theorem, which gives a purely graph- theoretic characterization of the skeletons of convex polyhedra: it states that the skeleton of every convex polyhedron is a 3-connected planar graph, and every 3-connected planar graph is the skeleton of some convex polyhedron.

In functional analysis and related areas of mathematics a dual topology is a locally convex topology on a dual pair, two vector spaces with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space. The different dual topologies for a given dual pair are characterized by the Mackey–Arens theorem. All locally convex topologies with their continuous dual are trivially a dual pair and the locally convex topology is a dual topology. Several topological properties depend only on the dual pair and not on the chosen dual topology and thus it is often possible to substitute a complicated dual topology by a simpler one.

Robert Connelly writes that, for a work describing significant developments in the theory of convex polyhedra that was however hard to access in the west, the English translation of Convex Polyhedra was long overdue. He calls the material on Alexandrov's uniqueness theorem "the star result in the book", and he writes that the book "had a great influence on countless Russian mathematicians". Nevertheless, he complains about the book's small number of exercises, and about an inconsistent level presentation that fails to distinguish important and basic results from specialized technicalities. Although intended for a broad mathematical audience, Convex Polyhedra assumes a significant level of background knowledge in material including topology, differential geometry, and linear algebra.

Pocchiola and Vegter (1996a,b,c) originally defined a pseudotriangle to be a simply-connected region of the plane bounded by three smooth convex curves that are tangent at their endpoints. However, subsequent work has settled on a broader definition that applies more generally to polygons as well as to regions bounded by smooth curves, and that allows nonzero angles at the three vertices. In this broader definition, a pseudotriangle is a simply-connected region of the plane, having three convex vertices. The three boundary curves connecting these three vertices must be convex, in the sense that any line segment connecting two points on the same boundary curve must lie entirely outside or on the boundary of the pseudotriangle.

In nonlinear optimization, quasiconvex programming studies iterative methods that converge to a minimum (if one exists) for quasiconvex functions. Quasiconvex programming is a generalization of convex programming.: Quasiconvex programming is used in the solution of "surrogate" dual problems, whose biduals provide quasiconvex closures of the primal problem, which therefore provide tighter bounds than do the convex closures provided by Lagrangian dual problems. In theory, quasiconvex programming and convex programming problems can be solved in reasonable amount of time, where the number of iterations grows like a polynomial in the dimension of the problem (and in the reciprocal of the approximation error tolerated); Kiwiel acknowledges that Yuri Nesterov first established that quasiconvex minimization problems can be solved efficiently.

Due to the circular path of a stream around a bend the surface of the water is slightly higher near the concave bank (the bank with the larger radius) than near the convex bank. This slight slope on the water surface of the stream causes a slightly greater water pressure on the floor of the stream near the concave bank than near the convex bank. This pressure gradient drives the slower boundary layer across the floor of the stream toward the convex bank. The pressure gradient is capable of driving the boundary layer up the shallow sloping floor of the point bar, causing sand, gravel and polished stones to be swept and rolled up- hill.

The spire shows moderately convex outlines:. The shell contains eight whorls. These are angular along the middle, with a moderately impressed suture. The aperture is very long and narrow.

The basal callus is white, yellow, reddish or black. The smooth surface is polished. The low spire is conical and acute. The 6 to 7 whorls are slightly convex.

Some very faint spirals show elsewhere in certain lights. The periphery is rounded. The base of the shell is convex and subperforate. The aperture is small and narrowly ovate.

The exterior of the outer lip is whitish. The protoconch is blunt. The 6 teleoconch whorls are not shouldered. They are slightly impressed on top and slightly convex below.

The columella is convex and its base is curved to the left.Kilburn, R.N. (1988) Turridae (Mollusca: Gastropoda) of southern Africa and Mozambique. Part 4. Subfamilies Drilliinae, Crassispirinae and Strictispirinae.

The Stiefel process is very similar to the Mannesmann process, except that the convex rollers are replaced with large conical disks. This allows for larger tubes to be formed.

The 11–15 mm. shell is globular with a depressed, low, conical spire. The whorls are convex, with quite deep sutures. The last whorl is angled on the periphery.

The brown, suborbicular shell is rather depressed. The conical spire is rather acute. The whorls enlarge rapidly. They are rather convex, concentrically striated with rather unequal acute spiral ridges.

All the osteoderms are highly convex and have radial grooves and ridges which are very deep or very high, hence the genus name. They also have many tiny pits.

The length of shell varies between 3.5 mm and 4.5 mm. The white shell is pellucid. The sculpture is decussated by microscopic striae. The teleoconch contains seven convex whorls.

The shell is internal, solid, ovate, convex above, flatly concave beneath, with a small apex, not coiled, lying on the right side as seen from about near the end.

These beetles can reach a length of about . Females are longer than males. They are very variable in size, punctation and color. Body is oval, weakly convex and shiny.

Rudolf Inzinger (5 April 1907 – 26 August 1980) was an Austrian mathematician who made contributions to differential geometry, the theory of convex bodies, and inverse problems for sound waves.

The shell has a maximum length of 18.5mm. The spire is high and acute, consisting of two and a half tightly wound, convex nuclear whorls and five postnuclear whorls.

The convex pentagons have all of their five angles smaller than 180° and no sides intersecting others. A common example of this type of pentagon is the regular pentagon.

The height of the shell attains 2½ mm, its largest diameter 4½ mm. The small shell has an ovate shape. Its back is convex. The shell is transversely striated.

The spire is conoid. The apex is generally eroded and orange-colored. The 6 whorls are convex. The body whorl is somewhat flattened or subconcave around the upper part.

The body whorl is angulate at the periphery, somewhat convex beneath. The aperture is subrhomboidal and smooth within. The columella is straightened in the middle. The umbilicus is narrow.

Every Fréchet lattice is a locally convex vector lattice. The set of all weak order units of a separable Fréchet lattice is a dense subset of its positive cone.

The planulate whorls with packed lirae, that are crenulate and transverse. The sutures are obliquely striate. The body whorl is subangulate. The base of the shell is slightly convex.

Hadwiger's conjecture is that parallelepipeds are the worst case for this problem, and that any other convex body may be covered by fewer than 2n smaller copies of itself.

The Lagrange multiplier method has several generalizations. In nonlinear programming there are several multiplier rules, e.g. the Carathéodory–John Multiplier Rule and the Convex Multiplier Rule, for inequality constraints.

The colour of the shell is pearly white. The suture is marked by the uppermost rib. The spire is turreted. The seven whorls are somewhat convex and gradually enlarging.

The shell has a diameter of 2.5 mm. The solid, yellowish white, subtranslucent shell has a depressed convex shape. It is more flattened below. The umbilicus is almost covered.

The spire is very depressedly conical. The apex is bluntly rounded, with a minute hyaline, depressed embryonic tip. The 4 whorls are barely convex. The suture is slightly impressed.

Head large. An indistinct lateral skin fold present. Forehead and back covered with large 16-18 rows of convex tubercles. Male are with 13-19 pre-ano-femoral pores.

The thin, white shell is transparent and polished. Its length measures 2.5 mm. The surface is microscopically longitudinally striated. The 4-5 whorls of the teleoconch are rather convex.

The first three categories are all convex, whereas Stewart toroids have polygonal-faced tunnels."Bonnie Stewarts Hohlkörper", by Christoph Pöppe, Spektrumdirekt (website of the German edition of Scientific American).

The shell has an elongate-conic shape. Its length measures .39 inch. The teleoconch contains 15 whorls that are moderately convex and are marked by a well impressed suture.

312.122 With extra resonator. 312.2 Half-tube zithers - The strings are stretched along the convex surface of a gutter. 312.21 Idiochord half-tube zithers. 312.22 Heterochord half-tube zithers.

For all convex pentagons, the sum of the squares of the diagonals is less than 3 times the sum of the squares of the sides.Inequalities proposed in “Crux Mathematicorum”, .

The city of Wintoncester, that fine old city, aforetime capital of Wessex, lay amidst its convex and concave downlands in all the brightness and warmth of a July morning.

Hinge and misclassification loss functions The simplest and most intuitive loss function for categorization is the misclassification loss, or 0–1 loss, which is 0 if f(x_i) = y_i and 1 if f(x_i) eq y_i, i.e. the Heaviside step function on -y_if(x_i). However, this loss function is not convex, which makes the regularization problem very difficult to minimize computationally. Therefore, we look for convex substitutes for the 0–1 loss.

Suppose that is a topological vector space and is a convex balanced and radial set. Then } is a neighborhood basis at the origin for some locally convex topology on . This TVS topology is given by the Minkowski functional formed by , , which is a seminorm on defined by . The topology is Hausdorff if and only if is a norm, or equivalently, if and only if } or equivalently, for which it suffices that be bounded in .

The shell size varies between 15 mm and 22 mm (Original description) The shell is thin, fragile, translucent, pale flesh-colored, moderately stout, with an acute, somewhat turreted spine. It contains 9 whorls. The 2½ whorls of the protoconch are nearly smooth, regular, convex and chestnut-colored. The subsequent whorls are shouldered, strongly convex in the middle, but with a smooth concave band below the suture, corresponding to the posterior notch in the outer lip.

A 2-dimensional uniform honeycomb is more often called a uniform tiling or uniform tessellation. Nearly all uniform tessellations can be generated by a Wythoff construction, and represented by a Coxeter–Dynkin diagram. The terminology for the convex uniform polytopes used in uniform polyhedron, uniform 4-polytope, uniform 5-polytope, uniform 6-polytope, uniform tiling, and convex uniform honeycomb articles were coined by Norman Johnson. Wythoffian tessellations can be defined by a vertex figure.

Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols The Schläfli symbol of a (convex) regular polygon with p edges is {p}. For example, a regular pentagon is represented by {5}. For (nonconvex) star polygons, the constructive notation {} is used, where p is the number of vertices and q - 1 is the number of vertexes skipped when drawing each edge of the star. For example, {} represents the pentagram.

Non-convex penalties - Penalties can be constructed such that A is constrained to be a graph Laplacian, or that A has low rank factorization. However these penalties are not convex, and the analysis of the barrier method proposed by Ciliberto et al. does not go through in these cases. Non-separable kernels - Separable kernels are limited, in particular they do not account for structures in the interaction space between the input and output domains jointly.

A multivariate polynomial is SOS-convex (or sum of squares convex) if its Hessian matrix H can be factored as H(x) = ST(x)S(x) where S is a matrix (possibly rectangular) which entries are polynomials in x. In other words, the Hessian matrix is a SOS matrix polynomial. An equivalent definition is that the form defined as g(x,y) = yTH(x)y is a sum of squares of forms.

In 2013 Amir Ali Ahmadi and Pablo Parrilo showed that every convex homogeneous polynomial in n variables and degree 2d is SOS-convex if and only if either (a) n = 2 or (b) 2d = 2 or (c) n = 3 and 2d = 4. Impressively, the same relation is valid for non-negative homogeneous polynomial in n variables and degree 2d that can be represented as sum of squares polynomials (See Hilbert's seventeenth problem).

Each central sphere can have up to twelve neighbors, and in a face-centered cubic lattice these take the positions of a cuboctahedron's vertices. In a hexagonal close- packed lattice they correspond to the corners of the triangular orthobicupola. In both cases the central sphere takes the position of the solid's center. Cuboctahedra appear as cells in three of the convex uniform honeycombs and in nine of the convex uniform 4-polytopes.

However, the strata are somewhat warped and so the escarpment's course is strongly convex to the north in the middle, gently convex to the south at either end. The escarpment begins where its determining limestone/dolosmite begins, in west- central New York. There, it separates the lowlands that separate Lake Ontario from Lake Erie. It curves to the northwest through the Ontario province to the island belt that divides the Georgian Bay from Lake Huron.

Juventus wordmark and icon introduced before the 2017–18 season In the past, the convex section of the emblem had a blue colour (another symbol of Turin) and it was concave in shape. The old French shield and the mural crown, also in the lower section of the emblem, had a considerably greater size. The two "Golden Stars for Sport Excellence" were located above the convex and concave section of Juventus' emblem.

Risk aversion (red) contrasted to risk neutrality (yellow) and risk loving (orange) in different settings. Left graph: A risk averse utility function is concave (from below), while a risk loving utility function is convex. Middle graph: In standard deviation-expected value space, risk averse indifference curves are upward sloped. Right graph: With fixed probabilities of two alternative states 1 and 2, risk averse indifference curves over pairs of state-contingent outcomes are convex.

The convex sets with the smallest possible Kovner–Besicovitch measure are the triangles, for which the measure is 2/3. The result that triangles are the minimizers of this measure is known as Kovner's theorem or the Kovner–Besicovitch theorem, and the inequality bounding the measure above 2/3 for all convex sets is the Kovner–Besicovitch inequality. The curve of constant width with the smallest possible Kovner–Besicovitch measure is the Reuleaux triangle.

The problem of finding all maximal points, sometimes called the problem of the maxima or maxima set problem, has been studied as a variant of the convex hull and orthogonal convex hull problems. It is equivalent to finding the Pareto frontier of a collection of points, and was called the floating-currency problem by Herbert Freeman based on an application involving comparing the relative wealth of individuals with different holdings of multiple currencies..

An antipodal pair of vertex and their supporting parallel lines. The rotating calipers method was first used in the dissertation of Michael Shamos in 1978. Shamos uses this method to generate all antipodal pairs of points on a convex polygon and to compute the diameter of a convex polygon in O(n) time. Godfried Toussaint coined the phrase "rotating calipers" and also demonstrated that the method was applicable in solving many other computational geometry problems.

Numerical modelling indicate that in periglacial settings broad low- angle convex hilltops can form in no less than millions of years. During the evolution of these slopes steeper initial slopes are calculated to result in the formation of numerous tors during the course of the lowering and broadening of the convex area. The presence of numerous tors would thus indicate that the original landscape was steeper and not flatter than present- day landscape.

Nonlinear Model Predictive Control, or NMPC, is a variant of model predictive control (MPC) that is characterized by the use of nonlinear system models in the prediction. As in linear MPC, NMPC requires the iterative solution of optimal control problems on a finite prediction horizon. While these problems are convex in linear MPC, in nonlinear MPC they are not necessarily convex anymore. This poses challenges for both NMPC stability theory and numerical solution.

Direct limits in the categories of topological spaces, topological vector spaces (TVSs), and Hausdorff locally convex TVSs are "poorly behaved". For instance, the direct limit of a sequence (i.e. indexed by the natural numbers) of locally convex nuclear Fréchet spaces may fail to be Hausdorff (in which case the direct limit does not exist in the category of Hausdorff TVSs). For this reason, only certain "well-behaved" direct systems are usually studied in functional analysis.

For the female, it is uniformly convex. The fifth ventrite in males has fine, shallow setae on an impression covering about one half its length, whereas the female's is uniformly convex. The male has an aedeagus whose internal sac has a pair of basal sclerites that are long, curved, and join at the anterior. The profemora are very small with a dull tooth, whereas the mesofemora and metafemora do not have a tooth.

Epinephelus maculatus has a body with a standard length which is 2.8 to 3.1 times its depth. The dorsal profile of the head is convex and the area between the eyes is either flat or marginally convex. The preopercle has a notch above the angle where there are enlarged serrations. The dorsal fin contains 11 spines and 15-17 soft rays while the anal fin has 3 spines and 8 soft rays.

The whitish gills are thick and closely spaced, sometimes developing pale pink tints. The cap is convex to broadly convex before flattening out in age, and reaches diameters between wide. The surface is dry to moist, smooth, and in maturity appears to be made of flattened fibers arranged radially. As the mushroom ages, the cap color changes from white to fuscous (dusky brownish grey) or brown, usually with olive, grayish or pale tan regions.

Polymake is software for the algorithmic treatment of convex polyhedra.Official Website Albeit primarily a tool to study the combinatorics and the geometry of convex polytopes and polyhedra, it is by now also capable of dealing with simplicial complexes, matroids, polyhedral fans, graphs, tropical objects, toric varieties and other objects. Polymake has been cited in over 100 recent articles indexed by Zentralblatt MATH as can be seen from its entry in the swMATH database.

The small fruit bodies typically grow in dense clusters. The cap is initially convex when young, later becoming convex to flattened or slightly depressed in the center, reaching a diameter of . The cap margin starts out rolled or curved inward, but straightens out as it matures. The cap surface ranges from dry to moist, smooth to covered with fine whitish hairs, and is mostly even with translucent radial grooves at the margin.

The gills are crowded, and whitish to pale yellow in color. The cap is wide, obtuse to convex, becoming broadly convex with a depressed center. The margin (cap edge) is rolled inward and bearded with coarse white hairs when young. The cap surface is dry and fibrillose except for the center, which is sticky and smooth when fresh, azonate, white to cream, becoming reddish-orange to vinaceous (red wine-colored) on the disc with age.

Within the genus Amanita, it is in the subgenus Lepidella, section Lepidella and subsection Solitariae. It is possibly the same species as Amanita farinacea, and if so, farinacea takes precedence. The fruit body has a white or cream cap, which is convex and rounded when young and opening out and flattening to flat-convex or flat to around 8 cm in diameter. It is covered in large irregular patches of the veil, also coloured cream.

The shell is rather depressed, with dome-shaped spire, the periphery mainly rounded but indistinctly subangular in front of the aperture, the base rather strongly convex. The umbilicus is small, widened in the last half-whorl, contained about 5½ times in the diameter of shell (in some examples smaller, 6 times or 7½ times in diameter). Opaque, cinnamon-brown, without much gloss, smoothish, with low growth-wrinkles. Whorls are strongly convex and increase slowly.

The lateral line is incomplete, unbranched, and midlateral. In most species the dorsal profile is straight, though it may be slightly convex from the head to the dorsal fin origin in some species. The ventral profile is slightly convex at the abdomen but is straight posteriorly. The caudal peduncle depth is approximately equal to its length in most species, though the depth is less than the length in C. roae and greater in C. orientale.

With equilibrium defined as ‘competitive equilibrium’, the first fundamental theorem can be proved even if indifference curves need not be convex: any competitive equilibrium is (globally) Pareto optimal. However the proof is no longer obvious, and the reader is referred to the article on Fundamental theorems of welfare economics. The same result would not have been considered to hold (with non- convex indifference curves) under the tangency definition of equilibrium. The point x of Fig.

The palm tree belt stretches along the Arvand River from Abadan south-east over a distance of about 40 km and is bounded in the interior by a road. The width of the belt varies from 2 to 6 km, and on average it is 4 km. The width is greater in the concave parts of the river bends and smaller in the convex parts. The convex parts have higher river levees and topography.

The oval grouper has a compressed, oval-shaped body and its depth is 2.0 to 2.8 times its standard length. It has an oblique mouth and the lower jaw projects beyond the upper jaw. The dorsal profile of the head is convex while the intraorbital area is rather wide and convex. The preopercle is not smoothly rounded, but is not sharply angled, and has fine serration on its margin which are enlarged at its angle.

On the other hand, Steve Zelditch proved that the answer to Kac's question is positive if one imposes restrictions to certain convex planar regions with analytic boundary. It is not known whether two non-convex analytic domains can have the same eigenvalues. It is known that the set of domains isospectral with a given one is compact in the C∞ topology. Moreover, the sphere (for instance) is spectrally rigid, by Cheng's eigenvalue comparison theorem.

The 14 whorls show a very regular increase. They are very slightly convex, sharply acute-angled at the carina. The base of the shell is flat at the outer edge and barely convex in the middle, with a slight dip in toward the edge of the umbilicus which is strongly defined. The suture is linear, defined by the white carinal fillet, and also on the lower whorls by being very slightly impressed.

If then this shows that is absorbing in so assume that . Note that , where , and that . Notice that the set , which is contained in , is a union of two line segments intersecting at the origin (with each containing the origin in an open sub- interval) so that its convex hull, which is contained in the convex set , clearly contains a quadrilateral having the origin in its interior. This shows that is absorbing in , as desired.

The separating axis theorem (SAT) says that: Two convex objects do not overlap if there exists a line (called axis) onto which the two objects' projections do not overlap. SAT suggests an algorithm for testing whether two convex solids intersect or not. Regardless of dimensionality, the separating axis is always a line. For example, in 3D, the space is separated by planes, but the separating axis is perpendicular to the separating plane.

The Graham scan algorithm for convex hulls Graham scan is a widely used and practical algorithm for convex hulls of two- dimensional point sets, based on sorting the points and then inserting them into the hull in sorted order. Graham published the algorithm in 1972. The biggest little polygon problem asks for the polygon of largest area for a given diameter. Surprisingly, as Graham observed, the answer is not always a regular polygon.

The Frank–Wolfe algorithm is an iterative first-order optimization algorithm for constrained convex optimization. Also known as the conditional gradient method, reduced gradient algorithm and the convex combination algorithm, the method was originally proposed by Marguerite Frank and Philip Wolfe in 1956. In each iteration, the Frank–Wolfe algorithm considers a linear approximation of the objective function, and moves towards a minimizer of this linear function (taken over the same domain).

Oloid structure. Showing the two 240 degrees circular sectors and the convex hull. The plane shape of a developed Oloid surface An oloid is a three- dimensional curved geometric object that was discovered by Paul Schatz in 1929. It is the convex hull of a skeletal frame made by placing two linked congruent circles in perpendicular planes, so that the center of each circle lies on the edge of the other circle.

474,475 Originally the little n-cubes operad or the little intervals operad (initially called little n-cubes PROPs) was defined by Michael Boardman and Rainer Vogt in a similar way, in terms of configurations of disjoint axis-aligned n-dimensional hypercubes (n-dimensional intervals) inside the unit hypercube. Later it was generalized by May to little convex bodies operad, and "little disks" is a case of "folklore" derived from the "little convex bodies".

Lordosis is historically defined as an abnormal inward curvature of the lumbar spine. However, the terms lordosis and lordotic are also used to refer to the normal inward curvature of the lumbar and cervical regions of the human spine.Medical Systems: A Body Systems Approach, 2005 Similarly, kyphosis historically refers to abnormal convex curvature of the spine. The normal outward (convex) curvature in the thoracic and sacral regions is also termed kyphosis or kyphotic.

In the theory of high-dimensional convex polytopes, a facet or side of a d-dimensional polytope is one of its (d − 1)-dimensional features, a ridge is a (d − 2)-dimensional feature, and a peak is a (d − 3)-dimensional feature. Thus, the edges of a polygon are its facets, the edges of a 3-dimensional convex polyhedron are its ridges, and the edges of a 4-dimensional polytope are its peaks..

Their mandibles each have five "teeth" and their antennal scapes are short and do not exceed the posterior margin of the head. Minor workers have rectangular heads with weakly convex posterior margins in full face view Major workers have reddish brown to blackish brown bodies. Their heads are proportionately larger and almost square with convex posterior margin in frontal view. Mandibles of major workers are large and triangular, with an acute apical "tooth".

E. gurnetensis queen The head of the E. gurnetensis is longer than wide, with slightly convex sides and an occipital margin that may be straight or poorly convex. The propodeum is smoothly rounded in profile, and the legs are short and thick. The males are smaller than the queens, with an average body length of . The heads have large, visible ocelli, while the petiole is slightly higher than long with a triangular outline.

For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. The negative of a quasiconvex function is said to be quasiconcave. All convex functions are also quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a generalization of convexity. Quasiconvexity and quasiconcavity extend to functions with multiple arguments the notion of unimodality of functions with a single real argument.

An affine transformation preserves: # collinearity between points: three or more points which lie on the same line (called collinear points) continue to be collinear after the transformation. # parallelism: two or more lines which are parallel, continue to be parallel after the transformation. # convexity of sets: a convex set continues to be convex after the transformation. Moreover, the extreme points of the original set are mapped to the extreme points of the transformed set.

This model shares the name with the convex great rhombicuboctahedron, also called the truncated cuboctahedron. An alternate name for this figure is quasirhombicuboctahedron. From that derives its Bowers acronym: querco.

The mine consists of an elongate, gall- like blotch in the deeper tissues of the leaf, rather inflated and showing only as a slightly convex swelling on both leaf surfaces.

For instance, if we abstract sets of couples (x, y) of real numbers by enclosing convex polyhedra, there is no optimal abstraction to the disc defined by x2+y2 ≤ 1.

The glossy white, shell is translucent. Its length measures 3.75 mm. The teleoconch contains seven convex whorls. Those of the spire and upper half of the body are longitudinally plicate.

The interstices are obliquely striate. The sutures are canaliculate. They are furnished with a series of granules above. The base of the shell is convex, furnished with concentric granulose cinguli.

In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces. The concept of uniform convexity was first introduced by James A. Clarkson in 1936.

In geometry, a Hanner polytope is a convex polytope constructed recursively by Cartesian product and polar dual operations. Hanner polytopes are named after Olof Hanner, who introduced them in 1956..

The form is oval. The back of the shell is quite convex. It is solid, but thinner than Haliotis rufescens. The outer surface has a uniform dull reddish-brown color.

Jingisukan is a Japanese grilled mutton dish prepared on a convex metal skillet or other grill. The dish is particularly popular on the northern island of Hokkaidō and in China.

The external surface of the parietal bone is convex, smooth, and marked near the center by an eminence, the parietal eminence (parietal tuber), which indicates the point where ossification commenced.

Each whorl has 5 elevated transverse lines. The angle of the body whorl is rounded. The base of the shell is .slightly convex and contains numerous transverse lines, mostly punctate.

It is also the only convex uniform tiling that can not be created as a Wythoff construction. It can be constructed as alternate layers of apeirogonal prisms and apeirogonal antiprisms.

Milgrom and Segal (2002) demonstrate that the generalized version of the envelope theorems can also be applied to convex programming, continuous optimization problems, saddle-point problems, and optimal stopping problems.