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Look up convex or convexity in Wiktionary, the free dictionary. Convex or convexity may refer to: Convex lens, in optics Convex set, containing the whole...

crescent shape, is not convex. The boundary of a convex set in the plane is always a convex curve. The intersection of all the convex sets that contain a...

In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or...

Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently...

In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined...

Look up plano-convex in Wiktionary, the free dictionary. Plano-convex may refer to: Plano-convex lens, in optics Plano-convex, a type of mudbrick used...

Algorithms that construct convex hulls of various objects have a broad range of applications in mathematics and computer science. In computational geometry...

mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also...

reflecting. Convex polyhedra are a well-defined class of polyhedra with several equivalent standard definitions. Every convex polyhedron is the convex hull of...

In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is...

the Latin name of the lentil (a seed of a lentil plant), because a double-convex lens is lentil-shaped. The lentil also gives its name to a geometric figure...

mathematics, a convex space (or barycentric algebra) is a space in which it is possible to take convex combinations of any finite set of points. A convex space...

Strictly convex may refer to: Strictly convex function, a function having the line between any two points above its graph Strictly convex polygon, a polygon...

In geometry, a convex polyhedron whose faces are regular polygons is known as a Johnson solid, or sometimes as a Johnson–Zalgaller solid. Some authors...

combinations with positive coefficients. It follows that convex cones are convex sets. The definition of a convex cone makes sense in a vector space over any ordered...

Convex analysis is the branch of mathematics that studies convex sets, convex functions, and their applications to optimization, functional analysis,...

In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are...

Examples of convex curves include the convex polygons, the boundaries of convex sets, and the graphs of convex functions. Important subclasses of convex curves...

logarithmically convex or superconvex if log ∘ f {\displaystyle {\log }\circ f} , the composition of the logarithm with f, is itself a convex function. Let...

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n {\displaystyle n} -dimensional...

theorem or the hyperplane separation theorem, and has numerous uses in convex geometry. The theorem is named for the mathematicians Hans Hahn and Stefan...

Euclidean space is said to be in convex position or convex independent if none of the points can be represented as a convex combination of the others. A finite...

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...

In economics, convex preferences are an individual's ordering of various outcomes, typically with regard to the amounts of various goods consumed, with...

boundary of the polygon does not cross itself. All convex polygons are simple. Concave: Non-convex and simple. There is at least one interior angle greater...

In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function f : R d → R {\displaystyle...

Convex Computer Corporation was a company that developed, manufactured and marketed vector minisupercomputers and supercomputers for small-to-medium-sized...

In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points...

In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas:...

The dynamic convex hull problem is a class of dynamic problems in computational geometry. The problem consists in the maintenance, i.e., keeping track...

In geometric graph theory, a convex embedding of a graph is an embedding of the graph into a Euclidean space, with its vertices represented as points and...

Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons and that is not a uniform polyhedron...

non-prismatic convex uniform 4-polytopes. There are two infinite sets of convex prismatic forms, along with 17 cases arising as prisms of the convex uniform...

of "convexity" on metric spaces. Karl Menger defined a metric space as convex if any "segment" joining two points in that space has other points in it...

Convex and Concave is a lithograph print by the Dutch artist M. C. Escher, first printed in March 1955. It depicts an ornate architectural structure with...

unless the objective function is convex in a minimization problem, there may be several local minima. In a convex problem, if there is a local minimum...

analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces...

The Shapley–Folkman lemma is a result in convex geometry that describes the Minkowski addition of sets in a vector space. The lemma may be intuitively...

authors have studied the computation of the volume of high-dimensional convex bodies, a problem that can also be used to model many other problems in...

In mathematics, a convex graph may be a convex bipartite graph a convex plane graph the graph of a convex function This disambiguation page lists articles...

geodesically convex subset of M. A function f : C → R {\displaystyle f:C\to \mathbf {R} } is said to be a (strictly) geodesically convex function if the...

A convex cap is a well defined structure in mathematics commonly used in convex geometry for approximating convex shapes. It is used in the construction...

is a mirror with a curved reflecting surface. The surface may be either convex (bulging outward) or concave (recessed inward). Most curved mirrors have...

rotational symmetry. Uniform polyhedra can be divided between convex forms with convex regular polygon faces and star forms. Star forms have either regular...

with convex sets of geometry, one uses order-convex instead of "convex". A convex sublattice of a lattice L is a sublattice of L that is also a convex set...

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...

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...

in a Convex Mirror may refer to: Self-Portrait in a Convex Mirror (Parmigianino), a c. 1524 painting by Parmigianino Self-Portrait in a Convex Mirror...

(convex, non-stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles. There are two objects, one convex and...

This definition is valid for any function, but most used for convex functions. A proper convex function is closed if and only if it is lower semi-continuous...

An integrally convex set is the discrete geometry analogue of the concept of convex set in geometry. A subset X of the integer grid Z n {\displaystyle...

complex (self-intersecting, or crossed). Simple quadrilaterals are either convex or concave. The interior angles of a simple (and planar) quadrilateral ABCD...

cubical cells, meeting at right angles. The tesseract is one of the six convex regular 4-polytopes. The tesseract is also called an 8-cell, C8, (regular)...

geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral...

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...

that is topologically equivalent to the dodecahedron. Fifteen types of convex pentagons are known to tile the plane monohedrally (i.e., with one type...

mathematics, a convex body in n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is a compact convex set with non-empty...

In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces. The concept of uniform convexity was...

strictly convex space is a normed vector space (X, || ||) for which the closed unit ball is a strictly convex set. Put another way, a strictly convex space...

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...

Carathéodory's theorem is a theorem in convex geometry. It states that if a point x {\displaystyle x} lies in the convex hull C o n v ( P ) {\displaystyle...

Convex Polytopes is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional convex polyhedra...

vertex. Many types of irregular octahedra also exist, including both convex and non-convex shapes. The regular octahedron has eight equilateral triangle sides...

on a convex subset of a real vector space, such that for any real number y, the set of points on which the function value is at most y is a convex set...

In the field of mathematics known as convex analysis, the indicator function of a set is a convex function that indicates the membership (or non-membership)...

In geometry, a set K ⊂ Rd is defined to be orthogonally convex if, for every line L that is parallel to one of standard basis vectors, the intersection...

usually considered to be a convex quadrilateral in Euclidean geometry, but there are also crossed cases. If shape ABCD is a convex trapezoid, then the ABDC...

polygons in two dimensions. There are six convex and ten star regular 4-polytopes, giving a total of sixteen. The convex regular 4-polytopes were first described...

In mathematics, particularly in functional analysis and convex analysis, a convex series is a series of the form ∑ i = 1 ∞ r i x i {\displaystyle \sum...

graph is said to be convex if all of its faces (including the outer face) are convex polygons. Not all planar graphs have a convex embedding (e.g. the...

particular the subfields of convex analysis and optimization, a proper convex function is an extended real-valued convex function with a non-empty domain...

or more points are vertices of the convex hull, any four such points can be chosen. If on the other hand, the convex hull has the form of a triangle with...

In the mathematical field of graph theory, a convex bipartite graph is a bipartite graph with specific properties. A bipartite graph ( U ∪ V , E ) {\displaystyle...

are seven topologically distinct convex hexahedra, one of which exists in two mirror image forms. Additional non-convex hexahedra exist, with their number...

convexity. The simplest convex deltahedron is the regular tetrahedron, a pyramid with four equilateral triangles. There are eight convex deltahedra, which can...

mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building...

and convex angle. Concave angles have slightly concave sides and open to the medial (inner) area of the tusk. Convex angles have somewhat convex sides...

compact convex sets in locally convex topological vector spaces (TVSs). Krein–Milman theorem—A compact convex subset of a Hausdorff locally convex topological...

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...

problems in mathematical programming can be formulated as problems on convex sets or convex bodies. Six kinds of problems are particularly important: optimization...

K-convex functions, first introduced by Scarf, are a special weakening of the concept of convex function which is crucial in the proof of the optimality...

In convex analysis and optimization, the normal cone to a set at a point is a convex cone consisting of vectors that make a non-acute angle with every...

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...

particularly if it is not convex. Every kite is an orthodiagonal quadrilateral (its diagonals are at right angles) and, when convex, a tangential quadrilateral...

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...

which the function value at any convex combination of elements in the domain is greater than or equal to that convex combination of those domain elements...

the convex cone of positive semi-definite matrices. This order is usually employed to generalize the definitions of monotone and concave/convex scalar...

Local convex hull (LoCoH) is a method for estimating size of the home range of an animal or a group of animals (e.g. a pack of wolves, a pride of lions...

The polynomially convex hull contains the holomorphically convex hull. The domain G {\displaystyle G} is called holomorphically convex if for every compact...

The Convex Mirror is a c 1916 oil with pencil on wood panel painting by Australian artist George Washington Lambert. The work depicts the interior of Belwethers...

economic phenomenon is convex when "intermediates (or combinations) are better than extremes". For example, an economic agent with convex preferences prefers...

either convex (bending outward) or concave (bending inward). The intersection of three disks forms a circular triangle whose sides are all convex. An example...

simple polygon (non-self-intersecting), regardless of whether it is convex or non-convex, this angle is called an internal angle (or interior angle) if a...

The convex-billed cowbird (Pandanaris convexa) is an extinct species of bird in the family Icteridae, described in 1947 by Alden H. Miller. It is the only...

: X → R {\displaystyle p\colon X\to \mathbb {R} } which is subadditive, convex, and satisfies p ( 0 ) ≤ 0 {\displaystyle p(0)\leq 0} is also positively...

of 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...

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 simple polygon...

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...

barrier methods or IPMs) are algorithms for solving linear and non-linear convex optimization problems. IPMs combine two advantages of previously-known algorithms:...

This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings. There are three regular and eight...

faces. Etymologically, "cuboid" means "like a cube", in the sense of a convex solid which can be transformed into a cube (by adjusting the lengths of...

equilateral (all sides have the same length). Regular polygons may be either convex or star. In the limit, a sequence of regular polygons with an increasing...

assumptions on the function f {\displaystyle f} (for example, f {\displaystyle f} convex and ∇ f {\displaystyle \nabla f} Lipschitz) and particular choices of η...

a polytope, but a diagram showing how the elements meet. The classical convex polytopes may be considered tessellations, or tilings, of spherical space...

In mathematics, the lower convex envelope f ˘ {\displaystyle {\breve {f}}} of a function f {\displaystyle f} defined on an interval [ a , b ] {\displaystyle...

In mathematics, Minkowski's theorem is the statement that every convex set in R n {\displaystyle \mathbb {R} ^{n}} which is symmetric with respect to the...

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...

onto convex sets (POCS), sometimes known as the alternating projection method, is a method to find a point in the intersection of two closed convex sets...

Tilings of the Euclidean plane by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of...

that point. Subderivatives arise in convex analysis, the study of convex functions, often in connection to convex optimization. Let f : I → R {\displaystyle...

A magnifying glass is a convex lens—usually mounted in a frame with a handle—that is used to produce a magnified image of an object. A magnifying glass...

polyhedral combinatorics (the study of faces of convex polyhedra), convex geometry (the study of convex sets, in particular combinatorics of their intersections)...

categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are not necessarily locally convex. Other well-known examples...

subsets, convex cones, and convex sets are generalizations of vector subspaces: a vector subspace is also an affine subspace, a convex cone, and a convex set...

three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120. Some dodecahedra...

polytope is a convex polytope whose vertices all have integer Cartesian coordinates. That is, it is a polytope that equals the convex hull of its integer...

down to a point if the sides aligned with every axis have length zero. A convex polygon is degenerate if at least two consecutive sides coincide at least...

number of convex corners and Y the number of concave corners. By the previous fact, X=Y+4. Let XX the number of convex corners followed by a convex corner...

space R n {\displaystyle \mathbb {R} ^{n}} and K {\displaystyle K} is a convex centrally symmetric body. Minkowski's theorem, sometimes called Minkowski's...

\mathbb {R} ^{n}} is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s 0 ∈ S {\displaystyle s_{0}\in...

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...

are convex. With this terminology, a convex polyhedron is the intersection of a finite number of halfspaces and is defined by its sides while a convex polytope...

(MILP) problems, as well as to solve general, not necessarily differentiable convex optimization problems. The use of cutting planes to solve MILP was introduced...

of two curves that, as a result of a point of inflection from concave to convex or vice versa, have ends of the overall curve that point in opposite directions...

algorithm for constrained convex optimization. Also known as the conditional gradient method, reduced gradient algorithm and the convex combination algorithm...

shape is allowed. Polyominoes are examples of tiles that are either convex of non-convex, for which various combinations, rotations, and reflections can be...

In Euclidean plane geometry, a rectangle is a rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular...

positive definite, the problem is a special case of the more general field of convex optimization. Quadratic programming is particularly simple when Q is positive...

general enough to be applied to other settings, for example, with other convex loss functions. Consider the setting of supervised learning with f {\displaystyle...

LSE and use this formula internally. LSE is convex but not strictly convex. We can define a strictly convex log-sum-exp type function by adding an extra...

difference bound matrix (DBM) is a data structure used to represent some convex polytopes called zones. This structure can be used to efficiently implement...

the perception of a concave mask of a face appears as a normal convex face. While a convex face will appear to look in a single direction, and the gaze...

containers. Conversely, a convex meniscus occurs when the adhesion energy is less than half the cohesion energy. For example, convex menisci occur between...

linear functions include the continuous piecewise linear functions and the convex piecewise linear functions. In general, for every n-dimensional continuous...

In the mathematical area of game theory and of convex optimization, a minimax theorem is a theorem that claims that max x ∈ X min y ∈ Y f ( x , y ) = min...

class of polyhedra with six quadrilateral faces. A rectangular cuboid is a convex polyhedron with six rectangle faces. The dihedral angles of a rectangular...

mathematics known as functional analysis, a reflexive space is a locally convex topological vector space for which the canonical evaluation map from X {\displaystyle...

In convex analysis and the calculus of variations, both branches of mathematics, a pseudoconvex function is a function that behaves like a convex function...

the candles (possibly omitted from the painting for lack of room). The convex mirror at the back, in a wooden frame with scenes of The Passion painted...

real-valued functions that are convex on a real variable. Specifically, if a real-valued multivariable function is convex on one of its independent real...

The regular icosahedron (or simply icosahedron) is a convex polyhedron that can be constructed from a pentagonal antiprism by attaching two pentagonal...

Self-Portrait in a Convex Mirror is a 1975 poetry collection by the American writer John Ashbery. The title, shared with its final poem, comes from the...

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...

In mathematics, the support function hA of a non-empty closed convex set A in R n {\displaystyle \mathbb {R} ^{n}} describes the (signed) distances of...

spheres determines a specific volume known as the convex hull of the packing, defined as the smallest convex set that includes all the spheres. There are many...

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

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...

groups are sometimes regarded as strongly geometric as well. Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues...

In geometry, a composite polyhedron is a convex polyhedron that produces two convex, regular-faced polyhedra when sliced by a plane.[dubious – discuss]...

topology and convex geometry about sequences of convex sets. Specifically, given a sequence { K n } {\displaystyle \{K_{n}\}} of convex sets contained...

\mathbf {x} \in \mathbf {X} } is the optimization variable chosen from a convex subset of R n {\displaystyle \mathbb {R} ^{n}} , f {\displaystyle f} is...

non-expansive maps is closed under convex combinations, but not compositions. This class includes proximal mappings of proper, convex, lower-semicontinuous functions...

equal circles arranged in a closed chain are sufficient to determine a convex equilateral pentagon. Each circle's center is one of four vertices of the...

analysis and related areas of mathematics a dual topology is a locally convex topology on a vector space that is induced by the continuous dual of the...

driver-side mirror to have a planar, convex, or aspheric surface; an aspheric section is often combined with a larger convex section, and the two sections are...

the convex relaxation of the primal problem: The convex relaxation is the problem arising replacing a non-convex feasible set with its closed convex hull...

special case for n = 4 is known as a tesseract. It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments...

A gömböc (Hungarian: [ˈɡømbøt͡s]) is any member of a class of convex, three-dimensional and homogeneous bodies that are mono-monostatic, meaning that they...

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. As a Catalan...

specifically in convex analysis, the convex compactification is a compactification which is simultaneously a convex subset in a locally convex space in functional...

{\displaystyle \kappa } copies of the intervals. The concepts of convex sets and convex components are used in a proof that every totally ordered set endowed...

point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa. For the graph of a function f of differentiability...

polyhedra. They may be obtained by stellating and faceting the regular convex dodecahedron and icosahedron, and differ from these in having regular pentagrammic...

3-space; similarly the 3D cube is related to the infinite 2D square tiling. Convex 4-polytopes can be cut and unfolded as nets in 3-space. A 4-polytope is...

non-positive curvature and by Alexander & Bishop (1990) for general locally convex metric spaces. The Cartan–Hadamard theorem in conventional Riemannian geometry...

symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its sides. Given its side length...

Generalizations of the Farkas' lemma are about the solvability theorem for convex inequalities, i.e., infinite system of linear inequalities. Farkas' lemma...

antiprisms, the convex polyhedrons as in 5 Platonic solids and 13 Archimedean solids—2 quasiregular and 11 semiregular— the non-convex star polyhedra as...

In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells...

used to solve non-differentiable convex optimization problems. Many interesting problems can be formulated as convex optimization problems of the form...

functional analysis, a locally convex vector lattice (LCVL) is a topological vector lattice that is also a locally convex space. LCVLs are important in...

connected to form a convex polyhedron called its convex hull. A compound is a faceting of its convex hull.[citation needed] Another convex polyhedron is formed...

vertex-transitive and constructed from uniform 4-polytope facets. The complete set of convex uniform 5-polytopes has not been determined, but many can be made as Wythoff...

commuting family of continuous affine self-mappings of a compact convex subset in a locally convex topological vector space has a common fixed point. This theorem...

need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk...

Brunn–Minkowski theorem (Hermann Brunn 1887; Hermann Minkowski 1896) applied to convex sets; the generalization to compact nonconvex sets stated here is due to...

geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron. There are many kinds of duality...

over the real or complex numbers that at the same time is also a (locally convex) Fréchet space. The multiplication operation ( a , b ) ↦ a ∗ b {\displaystyle...

convex preferences (that do not prefer extremes to in-between values) and convex budget sets and on producers with convex production sets; for convex...

National Book Critics Circle Award for his collection Self-Portrait in a Convex Mirror (1975). In 2007, he became the first living poet to be anthologized...

(Tychonoff) fixed-point theorem: Let V be a locally convex topological vector space. For any nonempty compact convex set X in V, any continuous function f : X →...

superadditivity. Convex games are totally balanced: The core of a convex game is non-empty, and since any subgame of a convex game is convex, the core of...

finitely many line segments. These polygons include as special cases the convex polygons, star-shaped polygons, and monotone polygons. The sum of external...

for boosting. Boosting algorithms can be based on convex or non-convex optimization algorithms. Convex algorithms, such as AdaBoost and LogitBoost, can...

forming convex hulls are commuting operations. If S {\textstyle S} is a convex set then μ S + λ S {\displaystyle \mu S+\lambda S} is also a convex set; furthermore...

linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half...

geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space. There are several rather similar...

or Delone triangulation of a set of points in the plane subdivides their convex hull into triangles whose circumcircles do not contain any of the points;...

Unsolved problem in mathematics Is there any three-dimensional convex body with lower packing density than the sphere? More unsolved problems in mathematics...

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...

algorithm used for clipping polygons. It works by extending each line of the convex clip polygon in turn and selecting only vertices from the subject polygon...

Convex Polyhedra is a book on the mathematics of convex polyhedra, written by Soviet mathematician Aleksandr Danilovich Aleksandrov, and originally published...

regular. An equilateral quadrilateral must be convex; this polygon is a rhombus (possibly a square). A convex equilateral pentagon can be described by two...

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...

In plane geometry, a lune (from Latin luna 'moon') is the concave-convex region bounded by two circular arcs. It has one boundary portion for which the...

bounding box of a point set is the same as the minimum bounding box of its convex hull, a fact which may be used heuristically to speed up computation. In...

Toronto International Film Festival and was bought for distribution by Convex Group, owners of the Flexplay disposable DVD technology. Flexplay disks...

of concave function, whereas a fillet on an exterior corner is a line of convex function (in these cases, fillets are typically referred to as rounds)....

in a projective space of dimension n. A convex basis of a polytope is the set of the vertices of its convex hull. A cone basis consists of one point...

to locally convex topological vector spaces, which may be of infinite dimension. It asserts that if K {\displaystyle K} is a nonempty convex closed subset...

reflected as if it came from the other. Diverging (negative) lenses and convex mirrors do not focus a collimated beam to a point. Instead, the focus is...

In geometry, the snub disphenoid is a convex polyhedron with 12 equilateral triangles as its faces. It is an example of deltahedron and Johnson solid....

case could first give me by its perfect apparition, when I was with two convexes trying experiments about the sun, the unexpected knowledge...if I .......

the ellipsoid method is an iterative method for minimizing convex functions over convex sets. The ellipsoid method generates a sequence of ellipsoids...

Euclidean geometry, an isosceles trapezoid or isosceles trapezium is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides...

spaces. A Fréchet space X {\displaystyle X} is defined to be a locally convex metrizable topological vector space (TVS) that is complete as a TVS, meaning...

projective plane. No heptahedra are regular. There are 34 topologically distinct convex heptahedra, excluding mirror images. (Two polyhedra are "topologically distinct"...

instead of the sublinear property,R is convex, then R is a set-valued convex risk measure. A lower semi-continuous convex risk measure ϱ {\displaystyle \varrho...

utility function. Theorem 1 (Varian): If the preferences of all agents are convex and strongly monotone, then PEEF allocations exist. Proof: The proof relies...

geometry Conical hull, in convex geometry Convex hull, in convex geometry Carathéodory's theorem (convex hull) Holomorphically convex hull, in complex analysis...

Antimatroids are equivalent, by complementation, to convex geometries, a combinatorial abstraction of convex sets in geometry. Antimatroids have been applied...

caveat: many terms in Riemannian and metric geometry, such as convex function, convex set and others, do not have exactly the same meaning as in general...

mathematics, the Chambolle–Pock algorithm is an algorithm used to solve convex optimization problems. It was introduced by Antonin Chambolle and Thomas...

geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular...

generalization of the concept of the convex hull, i.e. every convex hull is an alpha-shape but not every alpha shape is a convex hull. For each real number α...

measure of difference between two points, defined in terms of a strictly convex function; they form an important class of divergences. When the points are...

and four vertices. The tetrahedron is the simplest of all the ordinary convex polyhedra. The tetrahedron is the three-dimensional case of the more general...

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...

reflector is a combination of a primary concave mirror and a secondary convex mirror, often used in optical telescopes and radio antennas, the main characteristic...

Nesterov is a Russian mathematician, an internationally recognized expert in convex optimization, especially in the development of efficient algorithms and...

In mathematics, an extreme point of a convex set S {\displaystyle S} in a real or complex vector space or affine space is a point in S {\displaystyle S}...

foci (or focal points) of the lens. For a converging lens (for example a convex lens), the focal length is positive and is the distance at which a beam...

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...

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...

In geometry, the elongated square pyramid is a convex composite polyhedron constructed by attaching the square base of an equilateral square pyramid to...

geometry, a lens is a convex region bounded by two circular arcs joined to each other at their endpoints. In order for this shape to be convex, both arcs must...

360°, while interior angles in a triangle add up to 180°. However, on a convex polyhedron, the angles of the faces meeting at a vertex add up to less than...

been proposed that use regions that are general convex polygons, not necessarily rectangular. Convex treemaps were developed in several steps, each step...

In geometry, the gyroelongated pentagonal rotunda is a Johnson solid, a convex polyhedron with thirty equilateral triangles, six regular pentagons, and...

are smooth orbifolds whenever the target space is convex. A variety X {\displaystyle X} is called convex if the pullback of the tangent bundle to a stable...

curvature, combining the convex ovolo and concave cavetto. When the concave part is uppermost, it is called a cyma recta but if the convex portion is at the...

It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point...

In mathematics, Minkowski's first inequality for convex bodies is a geometrical result due to the German mathematician Hermann Minkowski. The inequality...

consider convex polygons, but star polygons, like the pentagram, when considered, can also be regular. They use the same vertices as the convex forms, but...

double lattice packing density of any convex set, and more generally for the optimal packing density of any convex set. Some 1000-kwacha coins from Zambia...

the primal–dual paradigm in the enhanced setting of KKT conditions and convex programs. Their algorithm is weakly-polynomial: it solves O ( ( n + m )...

the bitruncated cubic honeycomb, a convex uniform honeycomb formed by the truncated octahedron, a space-filling convex polyhedron with 6 square faces and...

and convex. It is found in the thumb, the thorax, the middle ear, and the heel. In a saddle joint, one bone surface is concave while another is convex. This...

of stochastic orders.[citation needed] Convex order is a special kind of variability order. Under the convex ordering, A {\displaystyle A} is less than...

self-concordant function, that is also a barrier function for a particular convex set. Self-concordant barriers are important ingredients in interior point...

mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to a tuple of convex bodies in R n {\displaystyle...

assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case. The word...

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...

sets of G {\displaystyle {\mathcal {G}}} is a method to define locally convex topologies on the vector spaces of a pairing. A pairing is a triple ( X...

object however same is possible for other types of mirror (concave and convex) but only for a specific conditions. However the focal length of a plane...

function. Both LogSumExp and softmax are used in machine learning. The convex conjugate (specifically, the Legendre transformation) of the softplus function...

concentrate on consumers with convex preferences and convex budget sets and on producers with convex production sets; for convex models, the predicted economic...

interpretations including in terms of geometry, Bayesian statistics and convex analysis. The LASSO is closely related to basis pursuit denoising. Lasso...

of convex optimization that studies problems consisting of minimizing a convex function over the intersection of an affine subspace and a convex cone...

across the river toward the convex bank. As it flows along the floor of the river, it sweeps loose material toward the convex bank. This flow of the boundary...

projective metric, is an explicitly defined distance function on a bounded convex subset of the n-dimensional Euclidean space Rn. It was introduced by David...

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...

contributions in this field. In 1995 Toponogov made the conjecture: On a complete convex surface S homeomorphic to a plane the following equality holds: inf p ∈...

4-dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells. Any convex polyhedron's surface has Euler characteristic V − E + F = 2 , {\displaystyle...

are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces...

In geometry, a Hanner polytope is a convex polytope constructed recursively by Cartesian product and polar dual operations. Hanner polytopes are named...

vertex arrangements of the convex regular polygons. In general, for any natural number n, there are n-pointed non-convex regular polygonal stars with...

shaped and polished, as opposed to faceted. The resulting form is usually a convex (rounded) obverse with a flat reverse. Cabochon was the default method of...

In six-dimensional geometry, a pentic 6-cube is a convex uniform 6-polytope. There are 8 pentic forms of the 6-cube. The pentic 6-cube, , has half of the...

candidates are both optimal. If, on the other hand, we choose the strictly convex cost function proportional to the square of Euclidean distance ( c ( x ...

into a corresponding convex hull, as demonstrated in the accompanying graphics. These illustrate the W5 sphere cluster, W5 convex hull, and two Waterman...

The pentagonal rotunda is a convex polyhedron with regular polygonal faces. These faces comprise ten equilateral triangles, six regular pentagons, and...

the problem has a convex objective function that is to be minimized, it will generally be easier to solve in the presence of a convex feasible set and...

been proposed to triangulate a polygon. It is trivial to triangulate any convex polygon in linear time into a fan triangulation, by adding diagonals from...

F} (follows by construction of the duality gap) F {\displaystyle F} is convex and lower semi-continuous (equivalent to the first point by the Fenchel–Moreau...

convex shape in the plane that can cover any shape of diameter one Mahler's conjecture on the product of the volumes of a centrally symmetric convex body...

contingent cone, and the Clarke tangent cone. These three cones coincide for a convex set, but they can differ on more general sets. Let A {\displaystyle A} be...

adjustable mirror is the cheval glass, which the user can tilt. Convex mirrors Convex mirrors provide a wider field of view than flat mirrors, and are...

piece from a figure, its area decreases but its perimeter may not. The convex hull of a figure may be visualized as the shape formed by a rubber band...

Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant...

during the Akkadian period. The Plano-convex building was a fortified construction built extensively with plano-convex bricks. It displayed the socio-economic...

and }}\qquad C_{r}(x):=\{z\in X\mid \|z-x\|\leq r\}.} Any such ball is a convex and bounded subset of X , {\displaystyle X,} but a compact ball/neighborhood...

polyhedral combinatorics, the Kleetope of a polyhedron or higher-dimensional convex polytope P is another polyhedron or polytope PK formed by replacing each...

The study of integer points in convex polyhedra is motivated by questions such as "how many nonnegative integer-valued solutions does a system of linear...

now known to be impossible. Squares can be inscribed in any smooth or convex curve, such as a circle or triangle, but it remains unsolved whether a square...

as the majorization inequality, is a theorem in elementary algebra for convex and concave real-valued functions, defined on an interval of the real line...

moderate to strong curve along its length. The cutting edge was only on the convex side of the blade, unlike the guisarme or bill. The fauchard was likely...

tumbler. One such theory is that the glass originally had a pointed or convex base and could not be set down without spilling. Another is that they had...

theorem about ovals states that the area cut off by a secant of a smooth convex oval is not an algebraic function of the secant. Isaac Newton stated it...

In the geometry of convex polyhedra, blooming or continuous blooming is a continuous three-dimensional motion of the surface of the polyhedron, cut to...

In four-dimensional geometry, the 24-cell is a convex regular 4-polytope, a four-dimensional analogue of a Platonic solid. It is named for the 24 octahedra...

f(t)} is convex, so is A ↦ Tr ⁡ f ( A ) {\displaystyle A\mapsto \operatorname {Tr} f(A)} on Hn, and it is strictly convex if f is strictly convex. See proof...

polygon into convex polygons, several objectives have been studied. The optimal convex partitioning problem is to partition a non-convex polygon into...

~f(a)+g(b).} This follows immediately from the definition of the convex conjugate. For a convex function f {\displaystyle f} this also follows from the Legendre...

edges are joined and which are separated. The edges that are cut from a convex polyhedron to form a net must form a spanning tree of the polyhedron, but...

In convex geometry, a zonoid is a type of centrally symmetric convex body. The zonoids have several definitions, equivalent up to translations of the...

the intersection of convex sets, and is a variant of the alternating projection method (also called the projections onto convex sets method). In its...

origin and every convex neighborhood of the origin contains a balanced convex neighborhood of the origin (even if the TVS is not locally convex). This neighborhood...

structure as a regular octahedron, but with dihedral angles that are non-convex along three disjoint edges. Because it has no interior diagonals, it cannot...

In eight-dimensional geometry, a stericated 8-simplex is a convex uniform 8-polytope with 4th order truncations (sterication) of the regular 8-simplex...

any continuous function f {\displaystyle f} mapping a nonempty compact convex set to itself, there is a point x 0 {\displaystyle x_{0}} such that f (...

In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not...

1\right\}.} Conversely: Any locally convex topological vector space has a local basis consisting of absolutely convex sets. A common method to construct...

optimality-criteria are convex (or concave) functions, and therefore optimal-designs are amenable to the mathematical theory of convex analysis and their computation...

simplest way of measuring the home range is to construct the smallest possible convex polygon around the data but this tends to overestimate the range. The best...

Convex analysis Convex cone Convex function Convex set Legendre–Fenchel transformation Convex minimization Fenchel's duality theorem Geometry Convex geometry...

is usually curved: the concave side is used as a wok or frying pan, the convex side for cooking flatbreads and pancakes. There are also flat tawas. The...

4-element orthoscopic eyepiece consists of a plano-convex singlet eye lens and a cemented convex-convex triplet field lens achromatic field lens. This gives...

Characterization of normable spaces Locally convex topological vector space – a vector space with a topology defined by convex open sets Space (mathematics) – mathematical...

Some fundamental problems of this type are: Convex hull: Given a set of points, find the smallest convex polyhedron/polygon containing all the points...

define locally convex polar topologies on the vector spaces in a dual pair, as the polar set of a bounded set is an absolutely convex and absorbing set...

The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygons and are vertex-transitive,[citation needed] although they...

Carathéodory's theorem in convex geometry states that if a point x {\displaystyle x} of R d {\displaystyle \mathbb {R} ^{d}} lies in the convex hull of a set P...

In mathematics, an associahedron Kn is an (n − 2)-dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening...

function. A related concept is convexification – converting a non-convex function to a convex function. It is especially important in economics and mathematical...

rhombic dodecahedron is a convex polyhedron with 12 congruent rhombi as its faces. The rhombic triacontahedron is a convex polyhedron with 30 golden rhombi...

the perimeter of the convex hull of S. So w is the diameter of a circle with the same perimeter as the convex hull. For convex bodies K in three dimensions...

is a mathematical theorem in the fields of mathematical statistics and convex geometry. The Gaussian correlation inequality states: Let μ {\displaystyle...

more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in...

Convex preferences relate to averages between two points on an indifference curve. It comes in two forms, weak and strong. In its weak form, convex preferences...

face and m the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra (the Platonic solids), and four regular star polyhedra...

{\displaystyle V} . While the overall problem is non-convex, each sub-problem is typically convex and can be solved efficiently. Jain, Netrapalli and Sanghavi...

mathematician who worked on convex geometry and reflection groups. He asked Shephard's problem on the volumes of projected convex bodies, posed another problem...

21 June 1936) is a Russian mathematician. He works in differential and convex geometry. Burago studied at Leningrad University, where he obtained his...

The stepped lens design also causes less heat buildup than does a plano-convex lens of the same angle. Theatrical Fresnel lanterns are typically made in...

also written (DF)-spaces are locally convex topological vector space having a property that is shared by locally convex metrizable topological vector spaces...

absorption property in function spaces as well as from compact convex co-convex subsets of a convex set. An example of cap sets comes from the card game Set...

{\textstyle \arctan \left({\sqrt {2}}\right)\approx 54.74^{\circ }} . A convex polyhedron in which all of the faces are regular polygons is called a Johnson...

equilateral and the base is a regular pentagon. Because this pyramid remains convex and all of its faces are regular polygons, it is classified as the second...

In mathematics, particularly, in asymptotic convex geometry, Milman's reverse Brunn–Minkowski inequality is a result due to Vitali Milman that provides...

give rise to a graph. This is due to Steinitz's theorem, stating that any convex polyhedron can be drawn into a planar graph that is 3-connected, meaning...

{\displaystyle L^{\infty }(\Omega )} if and only if f {\displaystyle f} is convex. Discontinuous linear functional Renardy, Michael & Rogers, Robert C. (2004)...

various restrictions on f-vectors of convex simplicial polytopes, to this more general setting. The face lattice of a convex polytope, consisting of its faces...

Dual polyhedron article.) The dual of a convex polyhedron is also a convex polyhedron. There are nine convex isotoxal polyhedra based on the Platonic...

In geometry, a Blind polytope is a convex polytope composed of regular polytope facets. The category was named after the German couple Gerd and Roswitha...

icosahedron, sometimes called Jessen's orthogonal icosahedron, is a non-convex polyhedron with the same numbers of vertices, edges, and faces as the regular...

pentagrams meeting at each vertex. It shares the same vertex arrangement as the convex regular icosahedron. It also shares the same edge arrangement with the great...

largest possible cube that can pass through a hole in a unit cube. Many other convex polyhedra, including all five Platonic solids, have been shown to have the...

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...

building. Other aids are dark glasses, or observing the sky reflected in a convex mirror or in a pool of water. Irisations are named after the Greek goddess...

never compact. However, a ball in a normed vector space will always be convex as a consequence of the triangle inequality. A subset of a metric space...

functions f : X → E {\displaystyle f\colon X\to E} with values in some locally convex space E {\displaystyle E} (1975 Rybakov, 1981 Blondia, 2015 Beckmann and...

(or nonahedron) is a polyhedron with nine faces. There are 2606 types of convex enneahedra, each having a different pattern of vertex, edge, and face connections...

functional and convex analysis, and related disciplines of mathematics, the polar set A ∘ {\displaystyle A^{\circ }} is a special convex set associated...

The quotient of a locally convex space by a closed subspace is again locally convex. Indeed, suppose that X is locally convex so that the topology on X...

regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes...

pinkish brown or cream cap is 3–11 centimetres (1–4+1⁄2 inches) in diameter, convex initially before flattening out with age. There is a slight umbo, and the...

terminology, a bounded polyhedron is called a polytope. Analytically, a convex polyhedron is expressed as the solution set for a system of linear inequalities...

sculptures of the eight winds on its frieze. A pulvinated frieze (or pulvino) is convex in section. Such friezes were typical of 17th-century Northern Mannerism...

In seven-dimensional geometry, a hexic 7-cube is a convex uniform 7-polytope, constructed from the uniform 7-demicube. There are 16 unique forms. Small...

pentagonal orthobirotunda (J42). A Johnson solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform...

theorem is a result in convex analysis named after French mathematician Jean-Jacques Moreau. It shows that sufficiently well-behaved convex functionals on Hilbert...

The ovolo or echinus is a convex decorative molding profile used in architectural ornamentation. Its profile is a quarter to a half of a more or less flattened...

vertex-transitive, and all facets are uniform 5-polytopes. The complete set of convex uniform 6-polytopes has not been determined, but most can be made as Wythoff...

dodecahedron, rhombo-hexagonal dodecahedron or hexarhombic dodecahedron is a convex dodecahedron with eight rhombic and four hexagonal faces. Given that a {\displaystyle...

The curvatures of the stomach are the long, convex, lateral surface, and the shorter, concave, medial surface of the stomach, which are referred to as...

of anisohedral convex polygons, showing that there were no anisohedral convex hexagons but being unable to show there were no such convex pentagons, while...

vertices, and nonregular hexagonal faces. There are 3 types of monohedral convex hexagonal tilings. They are all isohedral. Each has parametric variations...

{\displaystyle \psi } is c-convex iff ψ = φ c {\displaystyle \psi =\varphi ^{c}} for some φ {\displaystyle \varphi } . Like in the case of convex transformation,...

-Lipschitz gradient. When every f i {\displaystyle f_{i}} is convex the function is convex, and an ε {\displaystyle \varepsilon } -optimal point is reachable...

radial basis is sometimes used. Then a convex set is assigned to the parameter vector, and the size of the convex set is minimized such that every possible...

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...

efficiently solved by interior point methods. All linear programs and (convex) quadratic programs can be expressed as SDPs, and the sum of squares hierarchy...

distance algorithm is a method of determining the minimum distance between two convex sets, first published by Elmer G. Gilbert, Daniel W. Johnson, and S. Sathiya...

Safety Standards, Section 571.111 S5.4.2. Retrieved 8 March 2023. "Each convex mirror shall have permanently and indelibly marked at the lower edge of...

Slater condition) is a sufficient condition for strong duality to hold for a convex optimization problem, named after Morton L. Slater. Informally, Slater's...

route is likely to be obtained. Hill climbing finds optimal solutions for convex problems – for other problems it will find only local optima (solutions...

five and may have been responsible for the first known proof that no other convex regular polyhedra exist. Euclid completely mathematically described the...

triangles as faces. is the dual polyhedron to the cube and one of eight convex deltahedra. The stella octangula, or eight-pointed star, is the only stellation...

monomials. Geometric programming is closely related to convex optimization: any GP can be made convex by means of a change of variables. GPs have numerous...

has eight faces, twelve edges and six vertices. It is a convex polyhedron, and like any convex polyhedron, it has Euler's characteristic of 2, according...

regular-faced convex polyhedra, line 13. Linti (2013), p. 41. Shepherd (1954). Berman (1971), p. 336, See table IV, the Properties of regular-faced convex polyhedra...

dimensions, the 3-3 duoprism or triangular duoprism is a four-dimensional convex polytope. The duoprism is a 4-polytope that can be constructed using Cartesian...

five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle...

The convex-tailed horned toad (Boulenophrys caudoprocta) is a species of frog in the family Megophryidae, endemic to China, and is only known from the...

mushroom can grow to 20 centimetres (8 inches) high with a reddish-brown convex to flattening cap up to 30 cm (12 in) across, the size leading to another...

Kirkpatrick–Seidel algorithm is an algorithm designed for computing the convex hull of a set of points in the plane, offering a time complexity of O (...

the need for showing that arbitrary curves become convex and separately studying the behavior of convex curves. Their results can also be extended to settings...

constructed from a triangular tiling extruded into prisms. It is one of 28 convex uniform honeycombs. It consists of 1 + 6 + 1 = 8 edges meeting at a vertex...

non-perpendicularly. Depending on context, cone may refer more narrowly to either a convex cone or projective cone. Cones can be generalized to higher dimensions....

3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented...

for uniformly convex Banach spaces. It asserts that if K {\displaystyle K} is a nonempty convex closed bounded set in uniformly convex Banach space and...

concept from convex analysis is a convex function from Rn to real numbers, which is defined through an inequality between its value on a convex combination...

In four-dimensional geometry, a runcinated 24-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular 24-cell...

If the second derivative is positive, the quadratic approximation is a convex function of t {\displaystyle t} , and its minimum can be found by setting...

the Krein-Smulian theorem can refer to two theorems relating the closed convex hull and compactness in the weak topology. They are named after Mark Krein...

to wooden columns made from tree trunks would have been relatively easy. Convex fluting was probably intended to imitate plant forms. Minoan and Mycenaean...

points contained in the respective polygons. Such a duoprism is convex if both bases are convex, and is bounded by prismatic cells. Four-dimensional duoprisms...

In six-dimensional geometry, a steric 6-cube is a convex uniform 6-polytope. There are unique 4 steric forms of the 6-cube. Runcinated demihexeract Runcinated...

plan of an absolutely continuous probability measure is the gradient of a convex function. More precisely, if μ {\displaystyle \mu } and ν {\displaystyle...

Mathematics at the University of Vienna. Boț's research focuces on convex analysis, convex optimization, nonsmooth optimization, and monotone operators. His...

The medium-sized to large long-muzzled head has a straight or slightly convex profile with wide-set eyes. The croup is sloping, the haunches well-muscled...

role in convex analysis. Given a convex (extended real) function, the epigraph might not be closed. But the lower semicontinuous hull of a convex function...

on an eyeball retina (the camera and eye focus light through an internal convex lens). In simple terms, a real point image is formed when all the rays of...

Subgradient methods are convex optimization methods which use subderivatives. Originally developed by Naum Z. Shor and others in the 1960s and 1970s, subgradient...

antiprism between its two halves. A Johnson solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform...

In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {5,3,3}. It is also called...

winners). So they consider two subsets of convex preferences: homogenous quadratic preferences, and separable convex preferences. In the homogeneous quadratic...

an arbitrary quadrilateral form a parallelogram. If the quadrilateral is convex or concave (not complex), then the area of the parallelogram is half the...

It is also a canonical polyhedron. A Johnson solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform...

geometry, the gift wrapping algorithm is an algorithm for computing the convex hull of a given set of points. In the two-dimensional case the algorithm...

triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two...

vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected...

+ b) where a and b are the lengths of adjacent sides. Unlike any other convex polygon, a parallelogram cannot be inscribed in any triangle with less than...

\sum _{S}\alpha _{S}=1,\alpha _{S}\geq 0\right)} . The convex closure of any set function is convex over [ 0 , 1 ] n {\displaystyle [0,1]^{n}} . Consider...

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...

Similarly, kyphosis historically refers to abnormal convex curvature of the spine. The normal outward (convex) curvature in the thoracic and sacral regions...

of the box. It is sufficient to find the smallest enclosing box for the convex hull of the objects in question. It is straightforward to find the smallest...

{\boldsymbol {\psi }}.} If G ( . , s ) {\displaystyle G(.,{\boldsymbol {s}})} is convex for all s ∈ S ψ {\displaystyle {\boldsymbol {s}}\in {\boldsymbol {S}}_{\boldsymbol...

Some other names for specific types of raised pavement markers include convex vibration lines, Botts' dots, delineators, cat's eyes, road studs, or road...

polytope is projectively unique (or projectively stable) if it has a unique convex realization up to projective transformations. The study of projectively...

semidefinite, then the problem is convex. If these matrices are neither positive nor negative semidefinite, the problem is non-convex. If P1, ... ,Pm are all zero...

In geometry, a dodecahedral prism is a convex uniform 4-polytope. This 4-polytope has 14 polyhedral cells: 2 dodecahedra connected by 12 pentagonal prisms...

(ジンギスカン; "Genghis Khan") is a Japanese grilled mutton dish prepared on a convex metal skillet or other grill. It is often cooked alongside beansprouts,...

cap of Hebeloma velutipes is about 2-6 centimeters in diameter and can be convex or umbonate. Its color is variable, and it can be pale, white, or brown...

1007/978-3-642-30964-9. ISBN 978-3-642-30964-9. Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10...

concave bank (cut bank or river cliff) and deposits sediments on an inner, convex bank which is typically a point bar. The result of this coupled erosion...

problem), or convex (minimization problem) and the constraint set is convex, then the program is called convex and general methods from convex optimization...

3-space that preserve the tiling] is transitive on vertices). There are 28 convex examples in Euclidean 3-space, also called the Archimedean honeycombs. A...

A second-order cone program (SOCP) is a convex optimization problem of the form minimize   f T x   {\displaystyle \ f^{T}x\ } subject to ‖ A i x + b i...

is convex. For example, {3} is an equilateral triangle, {4} is a square, {5} a convex regular pentagon, etc. Regular star polygons are not convex, and...

New York metropolitan area. Half-moon cookies are slightly dome-shaped (convex), have frosting on a fluffy angel cake base, and are common in Central New...

In convex analysis, a non-negative function f : Rn → R+ is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it...

icosidodecahedron is a nonconvex uniform polyhedron, indexed as U45. Its convex hull is a nonuniform truncated icosidodecahedron. Cartesian coordinates...

shown in red, and the red dashed lines indicate their convex hull, which is the smallest convex polyhedron that contains all of these points. The blue...

of Engineering for contributions to engineering design and analysis via convex optimization. Boyd received an B.A. degree in mathematics, summa cum laude...

it is non-convex. Self-intersecting 5-polytopes are also known as star polytopes, from analogy with the star-like shapes of the non-convex Kepler-Poinsot...

The "Nimrud lens" of Assyrians manufacture, a rock crystal disk with a convex shape believed to be a burning or magnifying lens. 13th century: The increase...

polyhedral combinatorics (the study of faces of convex polyhedra), convex geometry (the study of convex sets, in particular combinatorics of their intersections)...

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...

In convex geometry and the geometry of convex polytopes, the Blaschke sum of two polytopes is a polytope that has a facet parallel to each facet of the...

position but reducing their size. The expansion of a regular convex polytope creates a uniform convex polytope. For polyhedra, an expanded polyhedron has all...

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)...

In the geometry of convex polytopes, the Minkowski problem for polytopes concerns the specification of the shape of a polytope by the directions and measures...

paintings, including the Circumcision of Jesus and his Self-portrait in a Convex Mirror, seeking patronage of the Medici pope, Clement VII. Vasari records...

In Euclidean geometry, an ex-tangential quadrilateral is a convex quadrilateral where the extensions of all four sides are tangent to a circle outside...

incorporation of convex hub 16-inch alloy wheels replacing the previous concave versions. Thus the original flat spoke "star" wheels became a convex design, in...

adjacent equatorial faces is not convex, and thus not a Johnson solid.) A Johnson solid is one of 92 strictly convex polyhedra that are composed of regular...

of all of these n − 1 {\displaystyle n-1} half-spaces, and hence it is a convex polygon. When two cells in the Voronoi diagram share a boundary, it is a...

largest function that is convex and that equals four times the variance for pure states. That is, it equals four times the convex roof of the variance F...

applied to solve the nonconvex problem with convex objective function, rank constraints and other convex constraints, and is thus suitable to solve our...

locally convex topological vector spaces is a natural topological vector space structure on their tensor product. Namely, given locally convex topological...

faces of a truncated dodecahedron. A Johnson solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform...

vertex of the convex hull is chosen, one can then apply the formula using the previous and next vertices, even if those are not on the convex hull, as there...

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...

k} points in convex position. However, some of these pairs of convex points could be blocked from visibility by points within the convex polygon they...

; Saks, Michael E. (1988), "Combinatorial representation and convex dimension of convex geometries", Order, 5 (1): 23–32, doi:10.1007/BF00143895, S2CID 119826035...

{x}})+C\|{\vec {x}}\|_{1}} where g {\displaystyle g} is a differentiable convex loss function. The method is an active-set type method: at each iterate...

together at the rim, leaving the inside hollow and creating the shape of a convex lens. The top ("Ding") side has a center 'note' hammered into it and seven...

A burning glass or burning lens is a large convex lens that can concentrate the Sun's rays onto a small area, heating up the area and thus resulting in...

general equilibrium model. It posits that under certain economic assumptions (convex preferences, perfect competition, and demand independence), there must be...

a polyhedron with six quadrilaterals (four-sided polygons). As for all convex polyhedra, the cube has Euler characteristic of 2, according to the formula...

closo-boranate ion [B11H11]2−. There are 107,854,282,197,058 topologically distinct convex octadecahedra, excluding mirror images, having at least 11 vertices. (Two...

focal lengths, apertures, and other properties. While in principle a simple convex lens will suffice, in practice a compound lens made up of a number of optical...

In convex geometry, a spectrahedron is a shape that can be represented as a linear matrix inequality. Alternatively, the set of n × n positive semidefinite...

adjacent equatorial faces is not convex, and thus not a Johnson solid.) A Johnson solid is one of 92 strictly convex polyhedra that are composed of regular...

"archetypes". The archetypes are convex combinations of observations chosen so that observations can be approximated by convex combinations of the archetypes...

and perimeter 2πw. The Minkowski sum acts linearly on the perimeters of convex bodies, so the perimeter of K must be half the perimeter of this disk, which...

pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable TVS. A pseudometric on a set X {\displaystyle X} is a map d :...

describing three-dimensional convex polyhedra in terms of the distances between points on their surfaces. It implies that convex polyhedra with distinct shapes...

four-dimensional geometry, a runcinated 120-cell (or runcinated 600-cell) is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the...

region. In some variations of the problem, the region is restricted to be convex. For example, a circular disk of radius 1/2 can accommodate any plane curve...

flammeus ("flaming"). The shape of the cap of B. rubroflammeus is convex to broadly convex, and reaches a diameter of 6–12 cm (2.4–4.7 in). The margin of...

this information is also encoded in a convex polytope, which creates a powerful connection of the subject with convex geometry. Familiar examples of toric...

Olshevsky nicknamed it the yog-sothoth (after the Cthulhu Mythos deity). Its convex hull is a nonuniform truncated dodecahedron. Let ξ = − 3 2 − 1 2 1 + 4 ϕ...

result of convex analysis that says that for every vector x {\displaystyle x} in a Hilbert space H {\displaystyle H} and every nonempty closed convex C ⊆ H...

combinations and hulls may be considered as convex combinations and convex hulls in the projective space. While the convex hull of a compact set is also a compact...

linear algebra, a power cone is a kind of a convex cone that is particularly important in modeling convex optimization problems. It is a generalization...

also convex. If S 1 , S 2 {\displaystyle S_{1},S_{2}} are convex and relatively open sets, then S 1 + S 2 {\displaystyle S_{1}+S_{2}} is convex and relatively...

found in patients with persistent facial erythema involving one or more convex surfaces of the face, with lesions that show granulomatous reaction histologically...

in the remaining cases. Chiliagon Myriagon 199,999 = 500,000 cases − 1 (convex) − 100,000 (multiples of 5) − 250,000 (multiples of 2) + 50,000 (multiples...

horizontally or row convex if its intersection with any horizontal line is convex. A polyomino is said to be convex if it is row and column convex. A polyomino...

isoquant, the higher the MRTS. Since MRTS must diminish, isoquants must be convex to their origin. Adding one input while holding the other constant eventually...

geometry Conformal geometry Constructive solid geometry Contact geometry Convex geometry Descriptive geometry Differential geometry Digital geometry Discrete...

The lateral, nearly circular, is concave from side to side, but slightly convex from before backward, especially at its posterior part, where it is prolonged...

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...

In convex analysis, Danskin's theorem is a theorem which provides information about the derivatives of a function of the form f ( x ) = max z ∈ Z ϕ ( x...

named after Hermann Minkowski, asks for the construction of a strictly convex compact surface S whose Gaussian curvature is specified. More precisely...

2024). "ServiceTitan Acquires Convex". FinSMEs. Retrieved November 26, 2025. "ServiceTitan to Acquire Software Firm Convex | ACHR News". www.achrnews.com...

Steinitz's theorem as being exactly the 3-vertex-connected planar graphs. Any convex polyhedron's surface has Euler characteristic V − E + F = 2 , {\displaystyle...

In geometry, the Bilinski dodecahedron is a convex polyhedron with twelve congruent golden rhombus faces. It has the same topology as the face-transitive...

{\displaystyle M} , the matroid polytope P M {\displaystyle P_{M}} is the convex hull of the indicator vectors of the bases of M {\displaystyle M} . Let...

Pitcher, with a ritual hunt scene, is a polished, black, convex, short-neck pitcher. It is kept in the History Museum of Armenia under the number 1919-1...

through the following diagram: f  convex ⟹ f  polyconvex ⟹ f  quasiconvex ⟹ f  rank-one convex {\displaystyle f{\text{ convex}}\implies f{\text{ polyconvex}}\implies...

Unsolved problem in mathematics Can every n {\displaystyle n} -dimensional convex body be covered by 2 n {\displaystyle 2^{n}} smaller copies of itself? More...

The convex-vented horned toad (Xenophrys pachyproctus), also known as the Gelin spadefoot toad or Huang's spadefoot toad, is a species of frog in the family...

Different convex sets yield different seminorms, and every asymmetric seminorm on R n {\displaystyle \mathbb {R} ^{n}} can be obtained from some convex set...

John Loudon McAdam c. 1820, in which crushed stone is placed in shallow, convex layers and compacted thoroughly. A binding layer of stone dust (crushed...

constraints are twice continuously differentiable, but not necessarily convex. SQP methods solve a sequence of optimization subproblems, each of which...

points in Euclidean space can be partitioned into subsets with intersecting convex hulls. Specifically, for any positive integers d , r {\displaystyle d,r}...

evolution. If the curve is non-convex, its total absolute curvature decreases monotonically, until it becomes convex. Once convex, the isoperimetric ratio of...

triangular bipyramid, it is a non-convex deltahedron because its faces are coplanar, thereby it is not strictly convex. Considering that each pair of its...

the property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator. Bornological...

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...

locally convex then we may add to this list: the gauge of D {\displaystyle D} is an infrabounded map; while if X {\displaystyle X} locally convex and Hausdorff...

dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular...

at New York University in New York City. His main research interests are convex geometry and its connections with analysis and information theory. Gaoyong...

generalized table of the most notable polyhedral stellations belonging to convex uniform polyhedra is provided, with complete sets of stellations for the...

geodesic circles). The result can also be extended to quasigeodesics on a convex polyhedron, and to closed geodesics of reversible Finsler 2-spheres. The...

A bullnose is a broad convex radius on construction materials such as wood, masonry, and ceramic tile. The term is also used for materials featuring such...

Restricting to the case of convex losses and coercive penalties Ciliberto et al. have shown that although Q is not convex jointly in C and A, a related...

face. They are sometimes also considered quasiregular. There are only two convex quasiregular polyhedra: the cuboctahedron and the icosidodecahedron. Their...

eight-pointed gold star with convex faceted rays and a relief image of a Chechen ornament. The star is bordered by a convex rim 0.5 mm wide. The distance...

{\displaystyle \theta } and C {\displaystyle C} is convex-valued, then C ∗ {\displaystyle C^{*}} is also convex-valued. If f {\displaystyle f} is strictly quasiconcave...

the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral combinatorics falls...

faces. It has 10p edges, and 4p vertices. There is an infinite series of convex uniform antiprismatic prisms, starting with the digonal antiprismatic prism...

pleurocystidia and cheilocystidia of this species. Cap: (1) 1.5 — 5 (8) cm across, convex to subumbonate, chestnut or orangish brown to yellowish brown to pearly...

Bourbaki (1950). 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...

sphenoid is convex, and divided by a transverse ridge, the infratemporal crest, into two portions. The superior or temporal portion, convex from above...

all edges. The dual of a convex polyhedron is also a convex polyhedron. The dual of a non-convex polyhedron is also a non-convex polyhedron. (By contraposition...