On a Class of Convex Sets and Functions

Given a convex subset C of ℝn, the set-valued mapping α↦α⋅C (where 0⋅C is, by convention, the recession cone of C) is increasing on ℝ+ if and only if C contains the origin, and decreasing on ℝ+ if and only if C is contained in its recession cone. This simple fact enables us to define a binary operation which combines a concave or convex function on ℝm with a convex subset of ℝn to produce a convex subset of ℝn+m. This binary operation is the set theoretic counterpart of a functional operation introduced by the author. In this paper, we present a detailed study of the class of convex subsets which are contained in their recession cones, and we establish some remarkable properties of our binary operation.