Convexity and almost convexity in groups

Witold Jarczyk. "Convexity and almost convexity in groups." Banach Center Publications 99.1 (2013): 55-76. <http://eudml.org/doc/281670>.

@article{WitoldJarczyk2013,
abstract = {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.},
author = {Witold Jarczyk},
journal = {Banach Center Publications},
keywords = {convex function; almost convex function; abelian group; locally compact group; extendibility of a function; set -ideal; difference equation; regularity of solution; integer programming; discrete convexity},
language = {eng},
number = {1},
pages = {55-76},
title = {Convexity and almost convexity in groups},
url = {http://eudml.org/doc/281670},
volume = {99},
year = {2013},
}

TY - JOUR
AU - Witold Jarczyk
TI - Convexity and almost convexity in groups
JO - Banach Center Publications
PY - 2013
VL - 99
IS - 1
SP - 55
EP - 76
AB - 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.
LA - eng
KW - convex function; almost convex function; abelian group; locally compact group; extendibility of a function; set -ideal; difference equation; regularity of solution; integer programming; discrete convexity
UR - http://eudml.org/doc/281670
ER -