Products of hyperbolic spaces

TL;DR

This paper introduces products of hyperbolic spaces that generalize known spaces and demonstrates that Reich's theorem applies to these new constructions, expanding the scope of fixed point theory.

Contribution

It defines and analyzes products of hyperbolic spaces, providing the first example where Reich's theorem holds outside CAT(0) spaces and normed spaces.

Findings

Products of hyperbolic spaces are well-behaved

Reich's theorem applies to these new spaces

First example outside CAT(0) and normed spaces

Abstract

The class of uniformly smooth hyperbolic spaces was recently introduced by the first author as a common generalization of both CAT(0) spaces and uniformly smooth Banach spaces, in a way that Reich's theorem on resolvent convergence could still be proven. We define products of such spaces, showing that they are reasonably well-behaved. In this manner, we provide the first example of a space for which Reich's theorem holds and which is neither a CAT(0) space, nor a convex subset of a normed space.