Convexity and almost convexity in groups

• Autorzy: Witold Jarczyk
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 39A06: Linear equations
• 22A10: Analysis on general topological groups
• 90C10: Integer programming
• 26A51: Convexity, generalizations
• 39B62: Functional inequalities, including subadditivity, convexity, etc.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2013
• Tom: 99
• Numer: 1
• Strony: 55-76

Daty

wydano

2013

Twórcy

autor

Witold Jarczyk

• Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, PL-65-516 Zielona Góra, Poland

Identyfikatory

DOI

10.4064/bc99-0-5