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Journal of Convex Analysis 14 (2007), No. 2, 239--247
Copyright Heldermann Verlag 2007

Generalized Convexity and Separation Theorems

Kazimierz Nikodem
Dept. of Mathematics, University of Bielsko-Biala, ul. Willowa 2, 43-309 Bielsko-Biala, Poland

Zsolt Páles
Institute of Mathematics, University of Debrecen, 4010 Debrecen Pf. 12, Hungary

Kakutani's classical theorem (stating that two disjoint convex sets can be separated by complementary convex sets) is extended to the setting when convexity is meant in the sense of Beckenbach. Then a characterization of pairs of functions that can be separated by a generalized convex function or by a function belonging to a two-parameter family (in the sense of Beckenbach) is presented. As consequences, stability results of the Hyers-Ulam-type are also obtained.

Keywords: Generalized convex sets, generalized convex functions, separation theorems.

MSC: 26A51, 52A01, 39B62

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