A real-valued function f defined on a convex subset D of some normed linear space X is said to be inner γ-convex w.r.t. some fixed roughness degree γ > 0 if there is a ν ∈ [0, 1] such that
holds for all x 0, x 1 ∈ D satisfying ∥x 0 − x 1∥ = ν γ and −(1/ν)x 0 + (1 + 1/ν)x 1 ∈ D. The requirement of this kind of roughly generalized convex functions is very weak; nevertheless, they also possess properties similar to those of convex functions relative to their supremum. For instance, if an inner γ-convex function defined on some bounded convex subset D of an inner product space attains its maximum, then it has maximizers at some strictly γ-extreme points of D. In this paper, some sufficient conditions and examples for γ-convex functions and several properties relative to the location of their maximizers are given.
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The financial support offered by Vietnam’s National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2014.28 is acknowledged.
Dedicated to Professor Nguyen Khoa Son on the occasion of his 65th birthday.
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Phu, H.X. Inner γ-Convex Functions in Normed Linear Spaces. Vietnam J. Math. 43, 487–500 (2015). https://doi.org/10.1007/s10013-015-0125-3
- Generalized convexity
- Rough convexity
- Inner γ-convex function
- γ-Extreme point
Mathematics Subject Classification (2010)