Generalized Convexity and Separation Theorems

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.