Convex Cone - Definition

Definition

A subset C of a vector space V is a convex cone if αx + βy belongs to C, for any positive scalars α, β, and any x, y in C.

The defining condition can be written more succinctly as "αC + βC = C" for any positive scalars α, β.

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.

The empty set, the space V, and 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).

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 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.

The intersection of two convex cones in the same vector space is again a convex cone, but their union may fail to be one. The class of convex cones is also closed under arbitrary linear maps. In particular, if C is a convex cone, so is its opposite -C; and C(-C) is the largest linear subspace contained in C.