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Some properties of globally \(\delta\)-convex functions. (English) Zbl 0839.90092
Summary: The notions of \(\delta\)-convex and midpoint \(\delta\)-convex functions were introduced by Hu, Klee, and Larman (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 \(\delta\)-convex and midpoint \(\delta\)-convex functions are investigated. Concretely, it is shown when they are bounded (from above or from below). For instance, a \(\delta\)-convex function defined on the entire real line is always locally bounded, and midpoint \(\delta\)-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) \(\delta\)-convex and midpoint \(\delta\)-convex functions on the real line.

MSC:
90C25 Convex programming
26B25 Convexity of real functions of several variables, generalizations
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