Fixed point theorems for metric spaces with a conical geodesic bicombing

TL;DR

This paper establishes new fixed point theorems for a broad class of metric spaces, including Banach and Busemann spaces, using barycenter constructions and invariant measures, and provides a counterexample of an isometry without fixed points.

Contribution

It introduces fixed point theorems for metric spaces with a conical geodesic bicombing, expanding applicability to spaces like Banach and Busemann spaces.

Findings

Fixed point theorems for metric spaces with conical geodesic bicombing

Construction of a Busemann space with an isometry lacking fixed points

Use of barycenter and invariant measure techniques

Abstract

We derive two fixed point theorems for a class of metric spaces that includes all Banach spaces and all complete Busemann spaces. We obtain our results by the use of a 1-Lipschitz barycenter construction and an existence result for invariant Radon probability measures. Furthermore, we construct a bounded complete Busemann space that admits an isometry without fixed points.