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

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