Journal of Convex Analysis Volume 14 (2007), No. 2, 239–247 Generalized Convexity and Separation Theorems Kazimierz Nikodem Department of Mathematics, University of Bielsko-Biala, ul. Willowa 2, 43-309 Bielsko-Biala, Poland knikodem@ath.bielsko.pl Zsolt P´ ales * Institute of Mathematics, University of Debrecen, 4010 Debrecen, Pf. 12, Hungary pales@math.klte.hu Received: January 10, 2005 Revised manuscript received: November 21, 2005 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 2000 Mathematics Subject Classification: Primary 26A51, 52A01, 39B62 1. Introduction Geometrically, the convexity of a function f : I → R means that, for any two distinct points on the graph of f , the segment joining these points lies above the corresponding part of the graph. In 1937 Beckenbach [2] generalized this concept by replacing the segments by graphs of continuous functions belonging to a certain two-parameter family F of functions (for the precise definition, see Section 3). The so obtained generalized convex functions have many properties known for classical convexity (cf., e.g., [2], [3], [16], [5], [4], [11], [17]). Using a similar idea, Krzyszkowski [10] introduced the notion of generalized convex sets and proved, among others, a Caratheodory-type result for them. The aim of this paper is to present further results on generalized convex sets and gener- alized convex functions. In Section 2, we prove a Kakutani-type separation theorem for generalized convex sets. A characterization of pairs of functions that can be separated by a generalized convex functions is given in Section 3. The problem of separation by functions belonging to F is discussed in Section 4. As corollaries, some Hyers–Ulam-type stability results related to generalized convexity are obtained. * The research of the second author has been supported by the OTKA grants T-043080, T-038072. ISSN 0944-6532 / $ 2.50 c  Heldermann Verlag