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Concept
Convex cone
Summary
In linear algebra, a cone—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under positive scalar multiplication; that is, C is a cone if implies for every positive scalar s.
When the scalars are real numbers, or belong to an ordered field, one generally calls a cone a subset of a vector space that is closed under multiplication by a positive scalar. In this context, a convex cone is a cone that is closed under addition, or, equivalently, a subset of a vector space that is closed under linear combinations with positive coefficients. It follows that convex cones are convex sets.
In this article, only the case of scalars in an ordered field is considered.
A subset C of a vector space V over an ordered field F is a cone (or sometimes called a linear cone) if for each x in C and positive scalar α in F, the product αx is in C. Note that some authors define cone with the scalar α ranging over all non-negative scalars (rather than all positive scalars, which does not include 0).
A cone C is a convex cone if αx + βy belongs to C, for any positive scalars α, β, and any x, y in C.
A cone C is convex if and only if C + C ⊆ C.
This 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. Also note that the scalars in the definition are positive meaning that the origin does not have to belong to C. Some authors use a definition that ensures the origin belongs to C. Because of the scaling parameters α and β, cones are infinite in extent and not bounded.
If C is a convex cone, then for any positive scalar α and any x in C the vector It follows that a convex cone C is a special case of a linear cone.
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 αC = C and C + C = C, for any positive scalar α.
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