Convex cone

In linear algebra, a cone—sometimes called a linear cone to distinguish it from other sorts of cones—is a subset of a real 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 ${\displaystyle s}$. This is a broad generalization of the standard cone in Euclidean space.

A convex cone is a cone that is also 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]

The definition of a convex cone makes sense in a vector space over any ordered field, although the field of real numbers is used most often.