Convergence of Functions and their Moreau Envelopes on Hadamard Spaces

TL;DR

This paper extends a key convergence equivalence from Hilbert spaces to Hadamard spaces, linking Mosco convergence of convex functions with the pointwise convergence of their Moreau envelopes, and explores related implications.

Contribution

It proves the converse of a known implication, establishing the equivalence of Mosco and Moreau envelope convergence in Hadamard spaces, and introduces a metric for function convergence.

Findings

Mosco convergence implies Moreau envelope convergence in Hadamard spaces.

The converse of this implication is now proven.

A complete metric on convex functions aligns with Mosco convergence.

Abstract

A well known result of H. Attouch states that the Mosco convergence of a sequence of proper convex lower semicontinuous functions defined on a Hilbert space is equivalent to the pointwise convergence of the associated Moreau envelopes. In the present paper we generalize this result to Hadamard spaces. More precisely, while it has already been known that the Mosco convergence of a sequence of convex lower semicontinuous functions on a Hadamard space implies the pointwise convergence of the corresponding Moreau envelopes, the converse implication was an open question. We now fill this gap. Our result has several consequences. It implies, for instance, the equivalence of the Mosco and Frol\'ik-Wijsman convergences of convex sets. As another application, we show that there exists a complete metric on the cone of proper convex lower semicontinuous functions on a separable Hadamard space such that a sequence of functions converges in this metric if and only if it converges in the sense of Mosco.