Solution of a problem of S. Marcus concerning Jensen-convex functions

Abstract

This paper is devoted to the problem of characterizing the class \( \cal S \) of the stationary sets for J-convex functions \( \Delta \to {\Bbb R} \), where \( \Delta \) is a convex open subset of \( {\Bbb R}^n \). We prove, among others, that a set T belongs to the class \( \cal S \) if and only if T satisfies two conditions: the closure of the convex hull of T in the relative topology is the whole set \( \Delta \), and each J-convex function bounded above on T is continuous.