Abstract

Beckenbach families are sets of functions possessing the two characteristic properties of Euclidean lines: their members are continuous and each distinct pairs of points of the plane can be interpolated by a unique member of the family. Applying Beckenbach families, the notion of (planar) convexity can be extended. Moreover, generalized convex functions can also be studied in this framework. The aim of this note is to prove the analogue of the Radon, Helly, Carathéodory and Minkowski Theorems in this generalized setting. The most important properties of generalized convex functions are presented, as well. As applications, some separation results are given.

Author information

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

Mihály Bessenyei

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

besse@science.unideb.hu

Bella Popovics

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

bellapopovics@gmail.com

Keywords

  • Beckenbach families
  • convex sets and functions
  • separation theorems

Mathematics Subject Classification

52A10; 26A51, 39B62, 52A40

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Suggested citation

M. Bessenyei, A. Konkoly, B. Popovics. “Convexity with Respect to Beckenbach Families.” Journal of Convex Analysis 24 (2017), No. 1, 75–92. https://doi.org/10.68381/jca24006

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Published by Heldermann Verlag, 2017. Rights now held by Banach Press LLC.