The fixed point property and unbounded sets in spaces of negative curvature

TL;DR

This paper investigates the relationship between geodesic boundedness and the fixed point property in complete CAT(-1) spaces, extending known results from special cases like Hilbert balls and R-trees.

Contribution

It provides an affirmative answer to whether geodesic boundedness characterizes fixed point properties in complete CAT(-1) spaces.

Findings

Geodesic boundedness is necessary and sufficient for fixed point property in these spaces.

Extends results from Hilbert balls and R-trees to general CAT(-1) spaces.

Clarifies the role of unbounded sets in fixed point theory.

Abstract

Motivated by the well-known cases of the real Hilbert ball and complete R-trees, being both particular cases of CAT(-1) spaces, we give an affirmative answer to the question of whether the geodesically boundedness property is a necessary and sufficient condition for a closed convex subset K of a complete CAT(k) space, with k < 0, to have the fixed point property for nonexpansive mappings