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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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##### References:
 [1] DOI: 10.1017/S0004972700004895 · Zbl 0452.90066 · doi:10.1017/S0004972700004895 [2] DOI: 10.1080/02331938308842832 · Zbl 0514.26003 · doi:10.1080/02331938308842832 [3] DOI: 10.1080/02331938508843063 · Zbl 0585.26008 · doi:10.1080/02331938508843063 [4] DOI: 10.1080/02331939208843838 · Zbl 0815.26004 · doi:10.1080/02331939208843838 [5] DOI: 10.1137/0327055 · Zbl 0686.52006 · doi:10.1137/0327055 [6] DOI: 10.1007/BF00940531 · Zbl 0679.90055 · doi:10.1007/BF00940531 [7] DOI: 10.1007/BF00939374 · Zbl 0792.90070 · doi:10.1007/BF00939374 [8] Komlosi S., Generalized Convexity (1994) [9] Martos B., Nonlinear Programming Theory and Methods [10] DOI: 10.1007/BF01195979 · Zbl 0798.49024 · doi:10.1007/BF01195979 [11] DOI: 10.1007/BF02193061 · Zbl 0831.90105 · doi:10.1007/BF02193061 [12] Phu H.X., Journal of Optimization theroy and Applications 85 (1994) [13] DOI: 10.1080/01630569408816600 · Zbl 0817.52008 · doi:10.1080/01630569408816600 [14] Phu H.X., Journal of Optimization Theory and Applications 15 (1994) [15] Roberts A.W., Convex Function (1973) [16] Rockafellar R.T., Convex Analysis (1970) · Zbl 0193.18401 · doi:10.1515/9781400873173 [17] Schaible S., On Generalized Concavity in Optimization and Economics (1981) [18] Söllner B., Eigenschaften $$\gamma$$-grobkonvexer Mengen und Funktionen
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