Abstract

Let A be a subset of an Abelian group G. We say that f from A to the reals R\mathbb{R} is convex if 2f(x) ≤ f(x+h) + f(x-h) holds for every x, h from G such that x, x+h, x-h are in A. We show that several classical theorems on convex functions defined on Rn\mathbb{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\mathbb{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

Author information

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Witold Jarczyk

Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland

w.jarczyk@wmie.uz.zgora.pl}

Miklós Laczkovich

Dept. of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/C, 1117 Budapest, Hungary
and: Department of Mathematics, University College London, Gower Street, London WC1E 6BT, England

laczk@cs.elte.hu

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Suggested citation

W. Jarczyk, M. Laczkovich. “Convexity on Abelian Groups.” Journal of Convex Analysis 16 (2009), No. 1, 33–48. https://doi.org/10.68381/jca16002

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Published by Heldermann Verlag, 2009. Rights now held by Banach Press LLC.