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...

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...

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

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

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

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...

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

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...

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...

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...

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

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...

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

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

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...

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

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...

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...

In mathematics, and particularly convex analysis, the lower convex envelope f ˘ {\displaystyle {\breve {f}}} of a real-valued function f {\displaystyle...

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

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...

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

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

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...

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

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...

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...

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...

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 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...

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...

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...

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...

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...

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

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...

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...

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...

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

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...

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

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...

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...

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...

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)...

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

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, an invariant convex cone is a closed convex cone in a Lie algebra of a connected Lie group that is invariant under inner automorphisms...

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...

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...

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...

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...

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...

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

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...

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

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

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...

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

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...

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...

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

infinitely many non-convex deltahedra. A convex polyhedron is made of the intersection of finitely many planes. Of the eight convex deltahedra, three are...

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

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

probability that four randomly chosen points in the Euclidean plane form a convex quadrilateral. Together with Buffon's needle problem, it has been called...

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 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...

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)...

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

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...

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...

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

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

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...

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...

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

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

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...

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

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

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

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...

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...

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...

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...

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...

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...

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...

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...

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

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...

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...

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

things, the geometrical foundation for generalizing the convex subdifferential to non-convex functions. Of particular note is also their role in generalizing...

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...

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

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...

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...

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

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

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

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

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...

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...

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...

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...

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...

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...

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...

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...

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

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...

In functional analysis, the class of B-convex spaces is a class of Banach space. The concept of B-convexity was defined and used to characterize Banach...

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

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

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...

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...

\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...

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

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

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...

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...

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

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

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

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...

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...

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...

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

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...

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

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

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...

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...

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...

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...

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

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

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...

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

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...

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...

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

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...

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

finite-dimensional objects, such as convex bodies and normed spaces, as the dimension tends to infinity. It is at the intersection of convex geometry and functional...

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

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...

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...

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...

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

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...

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

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...

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

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

where f is a bounded function and g is a convex function. MMO problems play a central role in game theory, convex optimization and online machine learning...

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

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

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

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...

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...

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...

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...

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

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...

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

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

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;...

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...

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

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...

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...

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...

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...

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...

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

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

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

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

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 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...

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...

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

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

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}...

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

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...

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

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

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...

(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 →...

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...

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...

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

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...

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

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 (...

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...

structure whose definable sets in the domain are just finite unions of convex sets. A linearly ordered structure, M, with language L including an ordering...

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...

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...

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...

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...

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...

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...

{\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...

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

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...

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

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

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...

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 ∈...

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...

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

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...

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

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...

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

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

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...

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

(1923–2004) was an American mathematician, specializing in the geometry of convex bodies. Born in Montana, Firey moved with his family to Seattle when he...

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

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

Each category contains methods designed for dealing with convex, non-smooth, and non-convex problems, each differing in hyper-parameter settings and other...

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

three-dimensional convex polyhedra and higher-dimensional convex polytopes. It states that, if one forms an undirected graph from the vertices and edges of a convex d-dimensional...

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...

numbers, Ehrhart's volume conjecture gives an upper bound on the volume of a convex body containing only one lattice point in its interior. It is a kind of...

or very slightly convex. The forewing upperside ground colour is light brown with a straight brown postmedian line, two slightly convex basal lines and...

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

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

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...

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...

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

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...

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...

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...

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...

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

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

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...

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...

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...

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

boundary. In most work on this problem, D {\displaystyle D} is additionally a convex set of nonzero area. When three of the placed points lie on a line, they...

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...

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...

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

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...

heptagonal antiprism. There are 387,591,510,244 topologically distinct convex hexadecahedra, excluding mirror images, having at least 10 vertices. (Two...

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

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...

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...

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

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

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...

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...

In mathematics, Tarski's plank problem is a question about coverings of convex regions in n-dimensional Euclidean space by "planks": regions between two...

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

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...

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...

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...

Capunti is a type of short convex oval pasta resembling an open empty pea pod. Media related to Capunti at Wikimedia Commons Italy portal Food portal List...

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...

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

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...

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

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...

substriatus is grayish or brownish in color and slimy when wet. It is conical to convex and about 3.8-5.2 centimeters in diameter. The stipe is about 5-12 centimeters...

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

: 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...

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

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...

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...

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

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...

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

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

\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...

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 six-dimensional geometry, a stericated 6-simplex is a convex uniform 6-polytope with 4th order truncations (sterication) of the regular 6-simplex. There...

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...

and 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...

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...

The Curmsun Disc is a purportedly Viking Age convex-concave gold disc that gained scholarly attention in 2014. The disc, which is apparently dedicated...

Gomphidius maculatus measures 2–9 cm in diameter and is initially convex, becoming plano-convex or flattened with maturity. Sometimes it may develop a slight...

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...

Argentine mathematician who works on optimization and analysis (particularly: convex analysis, functional analysis and non-smooth analysis). Currently, she is...

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

the study of complex vector spaces, a set S is said to be polynomially convex if, for every point x outside of S, there exists a polynomial whose complex...

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...

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...

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...

classified it as Conocybe affinis. Cap: (5) 7-12 (20)mm wide, semiglobose to convex or campanulate. The surface is uniformly light brown or tan but fades with...

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...

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...

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

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

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

The hexapentakis truncated icosahedron is a convex polyhedron constructed as an augmented truncated icosahedron. It is a geodesic polyhedron {3,5+}3,0...

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...

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

centimeters in diameter. It becomes browner in color as it gets older. It is convex to flat and sometimes depressed. The gills are pale to pinkish and adnate...

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...

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...

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 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...

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...

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

found on lawns. It is not edible. The cap is 1 to 4 cm across, conic to convex, hygrophanous with a brownish colour when moist and tannish when dry, often...

-hedron, the name becomes Hendecahedron. In all the convex hendecahedra, there are a total of 440,564 convex ones with distinct differences in topology. There...

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

quadrilateral has two diagonals, joining opposite pairs of vertices. For any convex polygon, all the diagonals are inside the polygon, but for re-entrant polygons...

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)...

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...

Equivalently, a line segment is the convex hull of two points. Thus, the line segment can be expressed as a convex combination of the segment's two end...

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...

Swedish mathematician who worked on complex analysis, continuous groups, convex sets, set-valued analysis, and game theory. From 1952, he was lektor (assistant...

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...

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...

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...

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...

Horses of the Alter Real strain are always brown. Members of the breed have convex facial profiles, heavy muscling, intelligent and willing natures, with agile...

imperforate, yellowish-white shell has a conical shape with a convex base, . The six convex whorls are separated by a deep suture, which is waved by the...

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

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

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...

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

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...

{\displaystyle J_{87}} There are 6,415,851,530,241 topologically distinct convex heptadecahedra, excluding mirror images, having at least 11 vertices. (Two...

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

{\displaystyle \mathbb {Z} ^{d},d\geq 0} . Affine monoids are closely connected to convex polyhedra, and their associated algebras are of much use in the algebraic...

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

(probably 'Webbiana'). Convex-leaved field elm, St James's Church, Charlotte Street, Leith Stems of same Foliage of same Convex-leaved field elm, East...

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...

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

In eight-dimensional geometry, a cantellated 8-simplex is a convex uniform 8-polytope, being a cantellation of the regular 8-simplex. There are six unique...

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...

on finite unions of convex bodies of R n . {\displaystyle \mathbb {R} ^{n}.} Other examples of valuations on finite unions of convex bodies of R n {\displaystyle...

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

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...

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...

(convex hull) - If a point x of Rd lies in the convex hull of a set P, there is a subset of P with d+1 or fewer points such that x lies in its convex hull...

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...

mutually tangent convex sets. Thus, it is a shape bounded by three inward-curved sides, formed by the boundaries of these three convex sets. A pseudotriangulation...

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...

about 8 mm. They are broadly oval, convex, light ferruginous. The antennae re testaceous. The thorax is uniformly convex and surrounded with a narrow black...

compound can be connected to form a convex polytope called the convex hull. The compound is a facetting of the convex hull. In 5-polytope compounds constructed...

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

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...

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 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...

geometry, a rhombicuboctahedral prism is a convex uniform polychoron (four-dimensional polytope). It is one of 18 convex uniform polyhedral prisms created by...

the trapezo-rhombic dodecahedron or rhombo-trapezoidal dodecahedron is a convex dodecahedron with 6 rhombic and 6 trapezoidal faces. It has D3h symmetry...

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

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...

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...

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...

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...

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...

(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...

\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...

Biconvex optimization is a generalization of convex optimization where the objective function and the constraint set can be biconvex. There are methods...

In seven-dimensional geometry, a pentic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 8 unique forms. Small cellated...

orientation. Every closed strictly convex curve is the envelope of its supporting lines. The astroid forms a non-convex hedgehog, and the deltoid curve forms...

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...

\operatorname {acl} A={\overline {A}}} for every finite-dimensional convex set A. Moreover, a convex set is algebraically closed if and only if its complement is...

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

functions of several complex variables. Suppose K {\displaystyle K} is a convex, compact, and symmetric subset of R n {\displaystyle \mathbb {R} ^{n}} ...

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...

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

In mathematics, the Milman–Pettis theorem states that every uniformly convex Banach space is reflexive. The theorem was proved independently by D. Milman...

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

{\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...

used, this construction produces a non-convex polyhedron with the same combinatorial structure as the convex triakis octahedron, a Catalan solid with...

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 ...

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

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...

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...

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 algorithmic treatment of convex polyhedra. Albeit primarily a tool to study the combinatorics and the geometry of convex polytopes and polyhedra, it...

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...

has contributed to the fields of mathematical economics, game theory, and convex analysis. Gale graduated with a Bachelor of Arts from Swarthmore College...

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

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

oracle) is a concept in the mathematical theory of convex optimization. It is a method to describe a convex set that is given as an input to an optimization...

Second World War. The mine's case is made of sheet steel, and has a slightly convex pressure plate on the top surface with a central fuze well. Two secondary...

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...

mismatch that requires correction. A good set of gauges will offer both convex and concave sections, and allow for their application in awkward locations...

The cap ranges from 2.5 to 7 cm (1–3 in) in diameter, and is initially convex before flattening. It has a slight boss that is mustard-, honey- or cream-yellow...

from a spherical to a convex shape. Smaller specimens tend to have more spherical pileus, while larger are more flattened convex shape, although both have...

{\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,...

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...

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...

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

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...

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...

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...

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...

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...

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

Austrian mathematician working in geometric number theory as well as in convex and discrete geometry. Gruber obtained his PhD at the University of Vienna...

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...

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...

in the center. It is initially convex, but already slightly undulatingly squashed while still young, becoming convex and depressed around the umbo, then...

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...

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...

Timothy M. Chan, is an optimal output-sensitive algorithm to compute the convex hull of a set P {\displaystyle P} of n {\displaystyle n} points, in 2- or...

measure is log-concave. The restriction of the Lebesgue measure to any convex set is also log-concave. By a theorem of Borell, a probability measure on...

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...

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...

are only eight different convex deltahedra, one of which is the gyroelongated square bipyramid. More generally, the convex polyhedron in which all faces...

problem whose first formulation required minimizing a non-convex function. For this non-convex minimization problem, Lemaréchal applied the theory of Lagrangian...

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

pale brown dapperling mushroom known from Australia. Cap: 1-2.5cm wide, convex and flattening with membranous flesh. The surface is cream or very light...

In five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex. There are unique 2 degrees...

portion is the larger, and is slightly concave transversely; the lateral is convex above, slightly concave below. The radial notch is a narrow, oblong, articular...

A scimitar (/ˈsɪmɪtər/ or /ˈsɪmɪtɑːr/) is a single-edged sword with a convex curved blade of about 75 to 90 cm (30 to 36 inches) associated with Middle...

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...

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...

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

In geometry, an octahedral prism is a convex uniform 4-polytope. This 4-polytope has 10 polyhedral cells: 2 octahedra connected by 8 triangular prisms...

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...

~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...

tridecahedron, and there are only five known regular convex polyhedra. There are 96,262,938 topologically distinct convex tridecahedra, excluding mirror images, having...

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

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...

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...

Repeatedly finding and removing a mouth from a non-convex polygon will eventually turn it into the convex hull of the initial polygon. This principle can...

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...

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

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

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...

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...

mushroom. Cap: 0.6-1.5cm wide and up to 1cm high starting campanulate to convex and not expanding much with age. The surface is hygrophanous and brown but...

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

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...

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...

Steinitz's theorem, the skeleton of a truncated icosahedron, like that of any convex polyhedron, can be represented as a polyhedral graph, meaning a planar graph...

often has spots. It is about 30-10 centimeters in diameter. At first, it is convex and has an inrolled margin. Then, it flattens out and often becomes depressed...

component S0 captures the moving objects in the foreground. Images of a convex, Lambertian surface under varying illuminations span a low-dimensional subspace...

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...

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...

characterizes the multisets of polygons that can form the faces of simple convex polyhedra. It states that, for given numbers of triangles, quadrilaterals...

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

pentagonal pyramids with equilateral triangles are used, the result is a non-convex deltahedron. Any such elevated dodecahedron has the same combinatorial structure...

optimization over particular geometries. We are given convex function f {\displaystyle f} to optimize over a convex set K ⊂ R n {\displaystyle K\subset \mathbb...

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...

area proof given above for the equilateral triangle generalizes to any (convex) equilateral polygon. In other words, if a polygon has all sides equal,...

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...

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

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...

Wiktionary, the free dictionary. A shield boss, or umbo (/ˈʌm.boʊ/), is a round, convex or conical piece of material at the centre of a shield. Shield bosses (or...

+ 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...

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

a result in geometry named after Shizuo Kakutani. It states that every convex body in 3-dimensional space has a circumscribed cube, i.e. a cube all of...

operations on polygons Convex hull Hyperplane arrangement Polygon decomposition Polygon triangulation Minimal convex decomposition Minimal convex cover problem...

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. Dürer's...

generalized by May to the little convex bodies operad, and "little disks" is a case of "folklore" derived from the "little convex bodies". In graph theory, rooted...

In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}. It is also known...

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...

reflector or an ellipsoidal reflector with a lens train containing two plano-convex lenses or a single step lens. The carbon-arc source was so bright that it...

The pentakis snub dodecahedron is a convex polyhedron with 140 triangular faces, 210 edges, and 72 vertices. It has chiral icosahedral symmetry. It comes...

"A Polynomial Number of Random Points Does Not Determine the Volume of a Convex Body". Discrete & Computational Geometry. 46 (1). Springer Science and Business...

In mathematics, the affine plank problem is an open question in convex geometry posed by Thøger Bang in 1951 as a strengthening of Tarski's plank problem...

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...

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...

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...

they used the scutum, which was larger. Originally, it was oblong and convex, but by the first century BC, it had developed into the rectangular, semi-cylindrical...

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

d-dimensional sphere. Some simplicial spheres arise as the boundaries of convex polytopes, however, in higher dimensions most simplicial spheres cannot...

In mathematics, a cyclic polytope, denoted C(n, d), is a convex polytope formed as a convex hull of n distinct points on a rational normal curve in Rd...

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...

curve or convex shape can be generalized to n dimension as a supporting hyperplane. If two bounded connected planar shapes have disjoint convex hulls that...

mathematics, symmetric cones, sometimes called domains of positivity, are open convex self-dual cones in Euclidean space which have a transitive group of symmetries...

In five-dimensional geometry, a stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-cube. There...

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...

whorls in total. The spire whorls are small and slightly convex, while the body whorl is large, convex, and gradually narrows anteriorly. The suture is superficial...

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

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...

classified it as Conocybe mesospora. Cap: 0.9-2.5 cm wide and semiglobate to convex or sometimes campanulate with a slight umbo. When young the centre is rust...

variational analysis is the combination and extension of methods from convex optimization and the classical calculus of variations to a more general...

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

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

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...

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...

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...

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