Journal of Convex Analysis 17 (2010), No. 2, 651--658
Copyright Heldermann Verlag 2010
On Maximal Domains for C-Convex Functions and Convex Extensions
Manfred Möller
School of Mathematics, University of the Witwatersrand, Wits 2050, South Africa
Manfred.Moller@wits.ac.za
Thabang M. J. Nthebe
School of Mathematics, University of the Witwatersrand, Wits 2050, South Africa
Thabang.Nthebe@wits.ac.za
[Abstract-pdf]
\newcommand\dom{\operatorname{dom}} Let $f$ be a real valued function with the domain $\dom(f)$ in some vector space $X$ and let $\mathfrak{C}$ be the collection of convex subsets of $X$. The following two questions are investigated; 1. Do there exist maximal convex restrictions $g$ of $f$ with $\dom(g) \in \mathfrak{C}$? 2. If $f$ is convex with $\dom(f)\in \mathfrak{C}$, do there exist maximal convex extension $g$ of $f$ with $\dom(g)\in \mathfrak{C}$? We will show that the answer to both questions is positive under a certain condition on $\mathfrak{C}$.
Keywords: Convex extension, C-convex, maximal set, CUP.
MSC: 32E20, 35E10, 26A51, 46A55, 46S40, 52A41
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