On Characterizations of (Almost) Strictly Convex Functions

TL;DR

This paper unifies and extends characterizations of (almost) strict convexity in convex analysis, linking subdifferential properties, monotonicity, and conjugate functions in finite and infinite-dimensional spaces.

Contribution

It generalizes existing characterizations of strict and almost strict convexity to Hilbert spaces and connects subdifferential monotonicity with convexity properties.

Findings

Subdifferentiable convex functions are strictly convex iff their subdifferential is strictly monotone.

Extensions of Rockafellar-Wets' characterizations to Hilbert spaces.

Results for paramonotone operators similar to convex function characterizations.

Abstract

In this paper, we unify and improve existing results on characterizing strict and almost stricty convex functions via subdifferential mapping, Moreau envelope, and proximal mappings. In particular, it is shown that if a convex function is subdifferentiable on its domain, then it is strictly convex if and only if its subdifferential is strictly monotone, equivalently, almost strictly monotone. Rockafellar-Wets' characterizations of almost strictly convex functions via almost differentiability of Fenchel conjugates and strict monotonicity of subdifferentials are extended from a finite-dimensional space to a Hilbert space. We also establish similar results for paramonotone operators.