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
knikodem@ath.bielsko.pl
Zsolt Páles
Institute of Mathematics, University of Debrecen, 4010 Debrecen Pf. 12, Hungary
pales@math.klte.hu
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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