Extension of continuous convex functions from subspaces II

Given Y a subspace of a topological vector space X, and an open convex set 0 is an element of A subset of X, we say that the couple (X, Y) has the CE(A)-property if each continuous convex function on A boolean AND Y admits a continuous convex extension defined on A. Using results from our previous paper, we study for given A the relation between the CE(A)-property and the CE(X)-property. As a corollary we obtain that (X, Y) has the CE(A)-property for each A, provided (X, Y) has the 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 CE(A)-property for sets A of special forms, or the 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 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.