Convex cone

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

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.^[1]

In this article, only the case of scalars in an ordered field is considered.