Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 22 (2015), No. 1, 101--116Copyright 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 [ Fulltext-pdf  (168  KB)] for subscribers only.