A characterization of essentially strictly convex functions on reflexive Banach spaces
Résumé
We call a function J : X → R ∪ {+∞} “adequate” whenever its tilted version by a continuous linear form x 7 → 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 Legendre-Fenchel conjugate J∗ of J, and the relationship between the essentially strict convexity of J and the Gˆateaux-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 express 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 Bregman-Tchebychev sets and functions for which some new results are established.
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