Geodesic
Convexity on Abelian Groups
Journal of convex analysis, Tome 16 (2009) no. 1, pp. 33-48 Cet article a éte moissonné depuis la source Heldermann Verlag

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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. 
@article{JCA_2009_16_1_JCA_2009_16_1_a1,
     author = {W. Jarczyk and M. Laczkovich},
     title = {Convexity on {Abelian} {Groups}},
     journal = {Journal of convex analysis},
     pages = {33--48},
     year = {2009},
     volume = {16},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JCA_2009_16_1_JCA_2009_16_1_a1/}
}
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W. Jarczyk; M. Laczkovich. Convexity on Abelian Groups. Journal of convex analysis, Tome 16 (2009) no. 1, pp. 33-48. http://geodesic.mathdoc.fr/item/JCA_2009_16_1_JCA_2009_16_1_a1/