C is a cone if x ∈ C {\displaystyle x\in C} implies s x ∈ C {\displaystyle sx\in C} for every positive scalar s. A cone need not be convex, or even...

a pyramid. Depending on the context, "cone" may also mean specifically a convex cone or a projective cone. Cones can also be generalized to higher dimensions...

Dual cone and polar cone are closely related concepts in convex analysis, a branch of mathematics. The dual cone C* of a subset C in a linear space X...

invariant convex cone is a closed convex cone in a Lie algebra of a connected Lie group that is invariant under inner automorphisms. The study of such cones was...

tangent cone, including the adjacent cone, Bouligand's contingent cone, and the Clarke tangent cone. These three cones coincide for a convex set, but...

a set of points is convex if it contains every line segment between two points in the set. Equivalently, a convex set or a convex region is a set that...

convex program can be presented in a conic form, which means minimizing a linear objective over the intersection of an affine plane and a convex cone:: 5...

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...

Thus we can take, for example, y1 = 1, y2 = −2. Consider the closed convex cone C ( A ) {\displaystyle C(\mathbf {A} )} spanned by the columns of A;...

cones (resp. convex cones) is again a cone (resp. convex cone); the same is true of the union of an increasing (under set inclusion) family of cones (resp...

bundles are described by a convex cone, and the possible contractions of the variety correspond to certain faces of the nef cone. In view of the correspondence...

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...

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...

semidefinite matrices forms a convex cone in Rn × n, and a spectrahedron is a shape that can be formed by intersecting this cone with an affine subspace. Spectrahedra...

cone is a kind of a convex cone that is particularly important in modeling convex optimization problems. It is a generalization of the quadratic cone:...

then the closed cone of curves is the closure of the positive cone. Otherwise, the closed cone of curves is the closed convex cone spanned by all elements...

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

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

Affine cone may refer to: Convex cone § Affine cone Cone (algebraic geometry) This disambiguation page lists articles associated with the same title....

hence they are members of the convex cone of nonnegative real numbers (whose nonzero members have reciprocals in this same cone). For several parameters,...

used to reduce affine geometry to linear algebra, and reduce convex bodies to convex cones. Explicitly, let S ⊂ R d {\displaystyle S\subset \mathbb {R}...

{\displaystyle \mathbb {Z} ^{n}} . A strongly convex rational polyhedral cone in N {\displaystyle N} is a convex cone (of the real vector space of N {\displaystyle...

positive definite matrices is an open convex cone, while the set of positive semi-definite matrices is a closed convex cone. Some authors use more general definitions...

In mathematics, especially convex analysis, the recession cone of a set A {\displaystyle A} is a cone containing all vectors such that A {\displaystyle...

prism), and a polyhedral cone (infinite cone) defined by three or more half-spaces passing through a common point. A convex polytope may be defined in...

hence they are members of the convex cone of nonnegative real numbers (whose nonzero members have reciprocals in this same cone). For several parameters,...

naturally from convex polyhedra, convex cones, and their associated discrete structures. Let C {\displaystyle C} be a rational convex cone in R n {\displaystyle...

realization of an abstract polytope is a convex cone of infinite dimension.: 127 : 229–230  The realization cone of the abstract apeirogon has uncountably...

matrix if the space of covariance matrices is viewed as an extrinsic convex cone in Rp×p; however, measured using the intrinsic geometry of positive-definite...

on an (abstract) convex cone of nonempty compact convex sets. The mapping τ {\displaystyle \tau } is an isometry between this cone, endowed with the...