
Journal of Convex Analysis 24 (2017), No. 1, 075092 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 [ Fulltextpdf (160 KB)] for subscribers only. 