On the global shape of convex functions on locally convex spaces

Abstract

In the recent paper [1] D. Azagra studies the global shape of continuous convex functions defined on a Banach space X. More precisely, when X is separable, it is shown that for every continuous convex function f:X→R there exist a unique closed linear subspace Y of X, a convex function h:X/Y→R with the property that limt→∞⁡h(u+tv)=∞ for all u,v∈X/Y, v≠0, and x⁎∈X⁎ such that f=h∘π+x⁎, where π:X→X/Y is the natural projection. Our aim is to characterize those proper lower semicontinuous convex functions defined on a locally convex space which have the above representation. In particular, we show that the continuity of the function f and the completeness of X can be removed from the hypothesis of Azagra's theorem. For achieving our goal we study general sublinear functions as well as recession functions associated to convex ones.