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Journal of Convex Analysis 16 (2009), No. 1, 033--048
Copyright Heldermann Verlag 2009

Convexity on Abelian Groups

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

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

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(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 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.

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