A characterization of essentially strictly convex functions on reflexive Banach spaces

摘要

We call a function J:X→R∪+∞ "adequate" whenever its version tilted by a continuous linear form x→J(x)- 〈~(x*),x〉 has a unique (global) minimizer on X, for appropriate ~(x*)∈~(X*). In this note we show that this induces the essentially strict convexity of J. The proof passes through the differentiability property of the LegendreFenchel conjugate ~(J*) of J, and the relationship between the essentially strict convexity of J and the Gteaux-differentiability of ~(J*). It also involves a recent result from the area of the (closed convex) relaxation of variational problems. As a by-product of the main result derived, we obtain an expression for the subdifferential of the (generalized) Asplund function associated with a couple of functions (f,h) with f∈Γ(X) cofinite and h:X→R∪+∞ weakly lower-semicontinuous. We do this in terms of (generalized) proximal set-valued mappings defined via (g,h). The theory is applied to BregmanTchebychev sets and functions for which some new results are established.