Porosity Results for Sets of Strict Contractions on Geodesic Metric Spaces

TL;DR

This paper studies the structure of nonexpansive mappings in various geodesic metric spaces, showing that strict contractions are rare and that typical mappings have Lipschitz constant one at most points.

Contribution

It demonstrates that strict contractions form a negligible, -porous subset in these spaces and that generic nonexpansive mappings are typically nonexpansive with Lipschitz constant one.

Findings

Strict contractions form a -porous subset of nonexpansive mappings.

A generic nonexpansive mapping has Lipschitz constant one at typical points.

Results apply to Banach, hyperbolic, and -( ext{ extsc{cat}}( ext{ extsc{kappa}})) spaces.

Abstract

We consider a large class of geodesic metric spaces, including Banach spaces, hyperbolic spaces and geodesic $\mathrm{CAT}(\kappa)$-spaces, and investigate the space of nonexpansive mappings on either a convex or a star-shaped subset in these settings. We prove that the strict contractions form a negligible subset of this space in the sense that they form a $\sigma$-porous subset. For separable metric spaces we show that a generic nonexpansive mapping has Lipschitz constant one at typical points of its domain. These results contain the case of nonexpansive self-mappings and the case of nonexpansive set-valued mappings as particular cases.