# Convexity and almost convexity in groups

Banach Center Publications (2013)

- Volume: 99, Issue: 1, page 55-76
- ISSN: 0137-6934

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topWitold 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 -

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