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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
∗This research was supported by the Deutsche Forschungsgemeinschaft. The author thanks Prof Dr. E. Zeidler, Prof. Dr. H.G Bock, and Dr. J Schlöder for their hospitality and valuable support during his visit in Leipzig and Heidelberg
∗This research was supported by the Deutsche Forschungsgemeinschaft. The author thanks Prof Dr. E. Zeidler, Prof. Dr. H.G Bock, and Dr. J Schlöder for their hospitality and valuable support during his visit in Leipzig and Heidelberg
Notes
∗This research was supported by the Deutsche Forschungsgemeinschaft. The author thanks Prof Dr. E. Zeidler, Prof. Dr. H.G Bock, and Dr. J Schlöder for their hospitality and valuable support during his visit in Leipzig and Heidelberg
