Strongly convex functions, Moreau envelopes and the generic nature of convex functions with strong minimizers

TL;DR

This paper introduces a new metric for convex functions using Moreau envelopes, demonstrating the density of strongly convex functions and the prevalence of convex functions with strong minima.

Contribution

It defines a complete metric space for convex functions and analyzes the topological properties of strongly convex functions and those with strong minima.

Findings

The set of strongly convex functions is dense but of first category.

The set of convex functions with strong minima is of second category.

Convergence in the metric corresponds to epi-convergence.

Abstract

In this work, using Moreau envelopes, we define a complete metric for the set of proper lower semicontinuous convex functions. Under this metric, the convergence of each sequence of convex functions is epi-convergence. We show that the set of strongly convex functions is dense but it is only of the first category. On the other hand, it is shown that the set of convex functions with strong minima is of the second category.