Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Kazimierz Nikodem

Dept. of Mathematics, University of Bielsko-Biała, ul. Willowa 2, 43-309 Bielsko-Biała, Poland

knikodem@ath.bielsko.pl

Zsolt Páles

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

pales@math.klte.hu

Keywords

  • Generalized convex sets
  • generalized convex functions
  • separation theorems

Mathematics Subject Classification

26A51, 52A01, 39B62

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Suggested citation

K. Nikodem, Z. Páles. “Generalized Convexity and Separation Theorems.” Journal of Convex Analysis 14 (2007), No. 2, 239–247. https://doi.org/10.68381/jca14017

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Published by Heldermann Verlag, 2007. Rights now held by Banach Press LLC.