Journal of Convex Analysis 22 (2015), No. 1, 101--116
Copyright Heldermann Verlag 2015
Extension of Continuous Convex Functions from Subspaces II
Carlo Alberto De Bernardi
Dipartimento di Matematica, Università degli Studi, Via C. Saldini 50, 20133 Milano, Italy
carloalberto.debernardi@gmail.com
Libor Veselý
Dipartimento di Matematica, Università degli Studi, Via C. Saldini 50, 20133 Milano, Italy
libor.vesely@unimi.it
[Abstract-pdf]
Given $Y$ a subspace of a topological vector space $X$, and an open convex set $0\in A\subset X$, we say that the couple $(X,Y)$ has the $\mathrm{CE}(A)$-property if each continuous convex function on $A\cap Y$ admits a continuous convex extension defined on $A$.\par Using results from our previous paper, we study for given $A$ the relation between the $\mathrm{CE}(A)$-property and the $\mathrm{CE}(X)$-property. As a corollary we obtain that $(X,Y)$ has the $\mathrm{CE}(A)$-property for each $A$, provided $(X,Y)$ has the $\mathrm{CE}(X)$-property and $Y$ is ``conditionally separable''. This applies, for instance, if $X$ is locally convex and conditionally separable. Other results concern either the $\mathrm{CE}(A)$-property for sets $A$ of special forms, or the $\mathrm{CE}(A)$-property for each $A$ where $X$ is a normed space with $X/Y$ separable.\par In the last section, we point out connections between the $\mathrm{CE}(X)$-property and extendability of certain continuous linear operators. This easily yields a generalization of an extension theorem of Rosenthal, and another result of the same type.
Keywords: Convex function, extension, topological vector space, normed linear space.
MSC: 52A41; 26B25, 47A99
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