Extension of Continuous Convex Functions from Subspaces I

Let $X$ be a topological vector space, $Y\subset X$ a subspace, and $A\subset X$ an open convex set containing $0$. We are interested in the extendability of a continuous convex function $f\colon A\cap Y\to\mathbb{R}$ to a continuous convex function $F\colon A\to\mathbb{R}$. We characterize such extendability: (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 [Boundedness, differentiability and extensions of convex functions, J. Convex Analysis 13 (2006) 587--602], and from another one by L. Zaj\'{\i}\v{c}ek and the second author [On extensions of d.c.\ functions and convex functions, J. Convex Analysis 17 (2010) 427--440]. We also show that if $X$ is locally convex and $X/Y$ is ``conditionally separable'', then the couple $(X,Y)$ satisfies the $\mathrm{CE}$-property, saying that the above extendability holds for $A=X$ and every $f$. It follows that every couple $(X,Y)$ has the $\mathrm{CE}$-property for the weak topology. \par We consider also a stronger $\mathrm{SCE}$-property saying that the above extendability is true for every $A$ and every $f$. A deeper study of the $\mathrm{SCE}$-property will appear in a subsequent paper.