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Convexity and almost convexity in groups

Tom 99 / 2013

Witold Jarczyk Banach Center Publications 99 (2013), 55-76 MSC: Primary 26 02, 39 02, 26A51, 39B62; Secondary 22A10, 39A40, 39A06, 90C10. DOI: 10.4064/bc99-0-5

Streszczenie

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.

Autorzy

  • Witold JarczykFaculty of Mathematics, Computer Science and Econometrics
    University of Zielona Góra
    Szafrana 4a
    PL-65-516 Zielona Góra, Poland
    e-mail

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