When is a Convex Cone the Cone of all the Half-Lines Contained in a Convex Set?

In this article we prove that every convex cone V of a real vector space X possessing an uncountable Hamel basis may be expressed as the cone of all the half-lines contained within some convex subset C of X (in other words, V is the infinity cone to C). This property does not hold for lower-dimensional vector spaces; more precisely, a convex cone V in a vector space X with a denumerable basis is the infinity cone to some convex subset of X if and only if V is the union of a countable ascending sequence of linearly closed cones, while a convex cone V in a finite-dimensional vector space X is the infinity cone to some convex subset of X if and only if V is Linearly closed.