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

Abstract

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