A characterization of essentially strictly convex functions on reflexive Banach spaces

We call a function J:X→R∪{+∞}J:X→R∪{+∞} “adequate” whenever its version tilted by a continuous linear form x↦J(x)−〈x∗,x〉x↦J(x)−〈x∗,x〉 has a unique (global) minimizer on XX, for appropriate x∗∈X∗x∗∈X∗. In this note we show that this induces the essentially strict convexity of JJ. The proof passes through the differentiability property of the Legendre–Fenchel conjugate J∗J∗ of JJ, and the relationship between the essentially strict convexity of JJ and the Gâteaux-differentiability of J∗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)(f,h) with f∈Γ(X) cofinite and h:X→R∪{+∞}h:X→R∪{+∞} weakly lower-semicontinuous. We do this in terms of (generalized) proximal set-valued mappings defined via (g,h)(g,h). The theory is applied to Bregman–Tchebychev sets and functions for which some new results are established.