On a Class of Convex Sets and Functions

Given a convex subset C of ⁿ, the set-valued mapping C (where 0C is, by convention, the recession cone of C) is increasing on ₊ if and only if C contains the origin, and decreasing on ₊ 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 ^m with a convex subset of ⁿ to produce a convex subset of ^n+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.Mathematics Subject Classifications (2000) 26A51, 26B25, 26E25.