
Journal of Convex Analysis 16 (2009), No. 1, 033048 Copyright Heldermann Verlag 2009 Convexity on Abelian Groups Witold Jarczyk Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65516 Zielona Góra, Poland w.jarczyk@wmie.uz.zgora.pl} Miklós Laczkovich Dept. of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/C, 1117 Budpest, Hungary and: Deptment of Mathematics, University College London, Gower Street, London WC1E 6BT, England laczk@cs.elte.hu Let A be a subset of an Abelian group G. We say that f from A to the reals R is convex if 2f(x) ≤ f(x+h) + f(xh) holds for every x, h from G such that x, x+h, xh are in A. We show that several classical theorems on convex functions defined on R^{n} can be proved in this general setting. We study extendibility of convex functions defined on subgroups of G. We show that a convex function need not have a convex extension, not even if it is defined on a subgroup of a linear space over the rationals Q. We give a sufficient condition of extendibility which is also necessary in groups divisible by 2. We also investigate the continuity and measurability of convex functions defined on topological Abelian groups. [ Fulltextpdf (171 KB)] for subscribers only. 