Proximal determination of convex functions

TL;DR

This paper explores how proximal mappings can be used to compare convex functions, establish their properties, and characterize Lipschitz continuity, offering new insights into convex analysis.

Contribution

It introduces comparison principles for convex functions via proximal mappings and characterizes Lipschitz continuity using the proximal operator.

Findings

Proximal mappings determine convex functions up to a constant.

Norm of the proximal operator characterizes the convex function.

A new characterization of Lipschitz continuity in terms of proximal operators.

Abstract

We provide comparison principles for convex functions through its proximal mappings. Consequently, we prove that the norm of the proximal operator determines a convex the function up to a constant. A new characterization of Lipschitzianity in terms of the proximal operator is given.