Fixed points and non-convex sets in ${\rm CAT}(0)$ spaces

Abstract

Dropping the condition of convexity on the domain of a nonexpansive mapping is a difficult and unusual task in metric fixed point theory. Hilbert geometry has been one of the most fruitful at which authors have succeeded to drop such condition. In this work we revisit some of the results in that direction to study their validity in ${\rm CAT}(0)$ spaces (geodesic spaces of global nonpositive curvature in the sense of Gromov). We show that, although the geometry of ${\rm CAT}(0)$ spaces resembles at certain points that one of Hilbert spaces, much more than the ${\rm CAT}(0)$ condition is required in order to obtain counterparts of fixed point results for non-convex sets in Hilbert spaces. We provide significant examples showing this fact and give positive results for spaces of constant negative curvature as well as $R$-trees.