The smallest convex extensions of a convex function

Abstract

Let C be a solid convex subset of a linear space X and be an algebraically-1.s.c. convex function. We prove the existence of a smallest convex extension of f to the whole space X, i.e. a minimal convex function such that for all x ∊ Cand provide an explicit formula for it. Some properties of the “smallest convex extension”-operator E are derived from this representation As application, we provide a simple, necessary and sufficient condition for a function , convex on C, to satisfy the convexity-lie inequalities for every integer n 2 and all (i.e. if just the average lies in the set C of convexity). Our condition, much weaker than convexity, is that f majorizes the smallest convex extension of fC , the restriction of f to C So we extend the results obtained in [1] under the essential additional assumption that C is algebraically open