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Journal of Convex Analysis 26 (2019), No. 3, 887--902
Copyright Heldermann Verlag 2019

A Sandwich Theorem for Generalized Convexity and Applications

Andrzej Olbrys
Institute of Mathematics, University of Silesia, 40-007 Katowice, Poland

In a recent paper we have introduced the concept of (ω, t)-convexity and ω-convexity as a natural generalization of the concept of usual convexity, strong-convexity, approximate-convexity, Wright-convexity and many others. The main result of this paper gives a necessary and sufficient condition on ω under which we can separate an (ω, t)-convex (ω-convex) function and (ω, t)-concave (ω-concave) function by an (ω, t)-affine map (ω-affine map). It turns out that in both cases ω has to satisfy some functional equation. We solve these functional equations in the most frequently appearing form of the map ω. Several applications of the main results are also given.

Keywords: Convexity, t-convexity, midpoint-convexity, separation theorem.

MSC: 26A51, 26B25, 39B62, 46C15, 52A41

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