A characterization of essentially strictly convex functions on reflexive Banach spaces

Abstract

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 Legendre–Fenchel conjugate J ∗ of J , and the relationship between the essentially strict convexity of J and the Gâteaux-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 Bregman–Tchebychev sets and functions for which some new results are established.