Extension of Continuous Convex Functions from Subspaces II

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 $A$. 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. 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.