Motivated by the well-known cases of the real Hilbert ball and complete a"e-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(kappa) space, with kappa < 0, to have the fixed point property for nonexpansive mappings.
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