Extension of Continuous Convex Functions from Subspaces I

Let X be a topological vector space, Y subset of X a subspace, and A subset of X an open convex set containing 0. We are interested in extendability of a continuous convex function f: A boolean AND Y -> R to a continuous convex function F: A -> R. We characterize such extendability being valid: (a) for a given f; (b) for every f. The case (b) for A = X generalizes results from a paper by J. Borwein, V. Montesinos and J. Vanderwerff, and from another one by L. Zajicek and the second-named author. We also show that if X is locally convex and X/Y is "conditionally separable" then the couple (X, Y) satisfies the CE-property, saying that the above extendability holds for A = X and every f. It follows that every couple (X, Y) has the CE-property for the weak topology.We consider also a stronger SCE-property saying that the above extendability is true for every A and every f. A deeper study of the SCE-property will appear in a subsequent paper.