Journal of Convex Analysis 24 (2017), No. 1, 075--092
Copyright Heldermann Verlag 2017
Convexity with Respect to Beckenbach Families
Mihály Bessenyei
Institute of Mathematics, University of Debrecen, Pf. 12, 4010 Debrecen, Hungary
besse@science.unideb.hu
Ágnes Konkoly
Institute of Mathematics, University of Debrecen, Pf. 12, 4010 Debrecen, Hungary
konkoly.agnes@freemail.hu
Bella Popovics
Institute of Mathematics, University of Debrecen, Pf. 12, 4010 Debrecen, Hungary
bellapopovics@gmail.com
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.
Keywords: Beckenbach families, convex sets and functions, separation theorems.
MSC: 52A10; 26A51, 39B62, 52A40
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