We give a review of results proved and published mostly in recent years, concerning real-valued convex functions as well as almost convex functions defined on a (not necessarily convex) subset of a group. Analogues of such classical results as the theorems of Jensen, Bernstein-Doetsch, Blumberg-Sierpiński, Ostrowski, and Mehdi are presented. A version of the Hahn-Banach theorem with a convex control function is proved, too. We also study some questions specific for the group setting, for instance the problem of the extendibility of a convex function from a subgroup to the whole group. What concerns almost convexity we present an abstract version of Kuczma's theorem. We sketch also some possible applications in improving regularity of solutions of a difference equation and in integer programming. The first appears, among others, in probability while determining weak generalized stable distributions, whereas the second is important in economics.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc99-0-5,
author = {Witold Jarczyk},
title = {Convexity and almost convexity in groups},
journal = {Banach Center Publications},
volume = {99},
year = {2013},
pages = {55-76},
zbl = {1281.26009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc99-0-5}
}
Witold Jarczyk. Convexity and almost convexity in groups. Banach Center Publications, Tome 99 (2013) pp. 55-76. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc99-0-5/