Convexity on Abelian Groups

Let A be a subset of an Abelian group G. We say that f from A to the reals

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

\mathbb{R}^n 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

\mathbb{Q} 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