Some properties of globally δ-convex functions∗

Abstract

The notions of δ-convex and midpoint δ-convex functions were introduced by Hu, Klee, and Larman (SIAM Journal on Control and Optimization, Vol. 27 1989). It is known that such functions have some important optimization properties: each r-local minimum is a global minimum, and if they assume their global maximum on a bounded convex domain of a Hilbert space then they do so at least at some r-extreme points of this domain. In this paper some analytical properties of δ-convex and midpoint δ-convex functions are investigated. Concretely, it is shown when they are bounded (from above or from below). For instance, δ-convex functions defined on the entire real line is always locally bounded, and midpoint δ-convex function on the real line is either locally bounded or totally unbounded. Further on, it is proved that there are totally discontinuous (i.e., nowhere differentiable) δ-convex and midpoint δ-convex functions on the real line