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In linear algebra, a convex cone is a subset of a vector space that is closed under linear combinations with positive coefficients. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...
A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ...
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ...
In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...
 A convex cone (light blue). Inside of it, the set of all points α x + β y with α>0 and β>0, given x and y in the cone (light red). The curves on the upper right symbolize that the regions are infinite in extent. Image File history File links This is a lossless scalable vector image. ...
Image File history File links This is a lossless scalable vector image. ...
Definition
A subset C of a vector space V is a convex cone if and only if αx + βy belongs to C, for any positive scalars α, β of V, and any x, y in C. It has been suggested that this article or section be merged into Logical biconditional. ...
The defining condition can be written more succinctly as "αC + βC = C for any positive scalars α, β of V.
The concept is meaningful for any vector space that allows the concept of "positive" scalar, such as spaces over the rational, algebraic, or (more commonly) the real numbers. In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...
In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
The empty set, the space V, an any linear subspace of V (including the trivial subspace {0}) are convex cones by this definition. Other examples are the set of all positive multiples of an arbitrary vector v of V, or the positive orthant of  (the set of all vectors whose coordinates are all positive). In linear algebra and related areas of mathematics, the null vector or zero vector in a vector space is the uniquely-determined vector, usually written 0, that is the identity element for vector addition. ...
In geometry of n dimensions, an orthant is one of the 2n parts of Euclidean space defined by given conditions on the Cartesian coordinates, that they are positive or negative. ...
A more general example is the set of all vectors λx such that λ is a positive scalar and x is an element of some convex set subset X of V. In particular, if V is a normed vector space, and X is an open (resp. closed) ball of V that does not contain 0, this construction gives an open (resp. closed) convex circular cone. Look up Convex set in Wiktionary, the free dictionary. ...
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ...
Balls are objects typically used in games. ...
Convex cones are closed under intersection, but not necessarily under union. They are also closed under arbitrary linear maps. In particular, if Cis a convex cone, so is its opposite -C; and C (-C) is the largest linear subspace contained in C. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
Convex cones are linear cones If C is a convex cone, then for any positive scalar α and any x in C the vector αx = (α/2)x + (α/2)x is in C. It follows that a convex cone C is a special case of a linear cone. In linear algebra, a (linear) cone is a subset of a vector space that is closed under multiplication by positive scalars. ...
Alternative definitions It follows from the above property that a convex cone can also be defined as a linear cone that is closed under convex combinations, or just under additions. More succinctly, a set C is a convex cone if and only if "λC = C and C + C = C, for any positive scalar α of V. A convex combination is a linear combination of data points (which can be vectors or scalars) where all coefficients are positive and sum up to 1. ...
3 + 2 = 5 with apples, a popular choice in textbooks[1] Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. ...
It follows also that one can replace the phrase "positive scalars α, β" in the definition of convex cone by "non-negative scalars α, β, not both zero".
Blunt and pointed cones According to the above definition, if C is a convex cone, then C {0} and C {0} are convex cones, too. A convex cone is said to be pointed or blunt depending on whether it includes the null vector 0 or not. Blunt cones can be excluded from the definition of convex cone by substituting "non-negative" for "positive" in the condition of α, β.
Half-spaces An hyperplane of V is a maximal proper linear subspace of V. An open (resp. closed) half-space of V is any subset H of V defined by the condition L(x) > 0 (resp. L(x) 0), where L is any linear function from V to its scalar field. The hyperplane defined by L(v) = 0 is the bounding hyperplane of H. A linear function is a mathematical function term of the form: f(x) = m x + c where c is a constant. ...
Half-spaces (open or closed) are convex cones. Moreover, any convex cone C that is not the whole space V must be contained in some closed half-space H of V. In fact, a topologically closed convex cone is the intersection of all closed half-spaces that contain it. The analogous result holds for any topologically open convex cone.
Salient convex cones and perfect half-spaces A convex cone is said to be flat if it contains some nonzero vector x and its opposite -x; and salient otherwise.
A blunt convex cone is necessarily salient, but the converse is not necessarily true. A convex cone C is salient if and only if C(-C) {0}; that is, if and only if C does not contain any non-trivial linear subspace of V.
A perfect half-space of V is defined recursively as follows: if V is zero-dimensional, then it is the set {0}, else it is any open half-space H of V, together with a perfect half-space of the bounding hyperplane of H.
Every perfect half-space is a salient convex cone; and, moreover, every salient convex cone is contained in a perfect half-space. In other words, the perfect half-spaces are the maximal salient convex cones (under the containment order). In fact, it can be proved that every pointed salient convex cone (independently of whether it is topologically open, closed, or mixed) is the intersection of all the perfect half-spaces that contain it. In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S. The term minimal element is defined dually. ...
Cross-sections and projections of a convex set
Flat section An affine hyperplane of V is any subset of V of the form v + H, where v is a vector of V and H is a (linear) hyperplane.
The following result follows from the property of containment by half-spaces. Let Q be an open half-space of V, and A = H + v where H is the bounding hyperplane of Q and v is any vector in Q. Let C be a linear cone contained in Q. Then C is a convex cone if and only the set C' = CA is a convex subset of A (i.e. a set closed under convex combinations). Look up Convex set in Wiktionary, the free dictionary. ...
A convex combination is a linear combination of data points (which can be vectors or scalars) where all coefficients are positive and sum up to 1. ...
Because of this result, all properties of convex sets of an affine space have an analog for the convex cones contained in a fixed open half-space.
Spherical section Given a norm |·| for V, we define the unit sphere of V as the set In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ...
A sphere is a perfectly symmetrical geometrical object. ...
 If the values of |·| are scalars of V, then a linear cone C of V is a convex cone if and only if its spherical section C' S (the set of its unit-norm vectors) is a convex subset of S, in the following sense: for any two vectors u, v in C' with u  -v, all the vectors in the shortest path from u to v in S are in C' . Look up cone in Wiktionary, the free dictionary. ...
In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...
In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. ...
Dual cone -
Let  be a convex cone in a real vector space V equipped with a scalar product. A dual cone to C is a set A set and its dual cone . A set and its polar cone . ...
 This is also a convex cone. If C is equal to its dual cone, C is called self-dual.
Partial order defined by a convex cone A pointed and salient convex cone cone C induces a partial ordering "≤" on V, defined so that x≤y if and only if y − x C. In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ...
Proper convex cone The term proper (convex) cone is variously defined, depending on the context. It often means a salient convex cone that is not contained in any hyperplane of V, possibly with other conditions such as topologically closed (and hence pointed), or topologically open (and hence blunt).
Examples of convex cones Given a closed, convex subset K of V, the normal cone to the set K at the point x in K is given by  Given a closed, convex subset K of V, the tangent cone (or contingent cone) to the set K at the point x is given by  Both the normal and tangent cone have the property of being closed and convex. They are important concepts in the fields of convex optimization, variational inequalities and projected dynamical systems. Convex optimization is a subfield of mathematical optimization. ...
Variational inequality is a mathematical theory which attempts to serve as a methodology for the study of equilibrium problems. ...
Projected dynamical systems is a mathematical theory investigating the behaviour of dynamical systems where solutions are restricted to a constraint set. ...
See also Look up cone in Wiktionary, the free dictionary. ...
This article is about the geometric object, for other uses see Cone. ...
In topology, especially algebraic topology, the cone CX of a topological space X is the quotient space: of the product of X with the unit interval I = [0, 1]. Intuitively we make X into a cylinder and collapse one end of the cylinder to a point. ...
In linear algebra, a (linear) cone is a subset of a vector space that is closed under multiplication by positive scalars. ...
External links - Jon Dattorro, Convex Optimization & Euclidean Distance Geometry
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