On a theorem about Mosco convergence in Hadamard spaces

TL;DR

This work addresses when convergence of proximal mappings $J^n_{\lambda}$ in Hadamard spaces implies Mosco convergence of a sequence of closed convex functions. By introducing the notion of asymptotically bounded slope and a normalization condition, the authors establish a converse to Attouch’s-type results: if $f^n\to f$ pointwise, the slopes are controlled in the limit, and $J^n_{\lambda}x\to J_{\lambda}x$, then $f^n_{\lambda}(x)\to f_{\lambda}(x)$ and, under these conditions, Mosco convergence follows. The results complete the equivalence cycle among Mosco convergence, Moreau envelopes, and proximal mappings in Hadamard spaces, mirroring Attouch’s theorem in smooth Banach spaces, and clarifying when proximal-map convergence suffices. A normalization condition is shown to be satisfiable under Mosco convergence, linking slope, function values, and proximal structure in a unified framework.

Abstract

Let $(f^n),f$ be a sequence of proper closed convex functions defined on a Hadamard space. We show that the convergence of proximal mappings $J^n_λx$ to $J_λx$, under certain additional conditions, imply Mosco convergence of $f^n$ to $f$. This result is a converse to a theorem of Bacak about Mosco convergence in Hadamard spaces.

Figures (1)

• Figure 1: Geodesic triangle (left) and its comparison triangle (right)

Theorems & Definitions (32)

• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 1
• Theorem 2
• Theorem 3
• Definition 1
• Lemma 1
• Lemma 2