Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings

It is shown that if the modulus ΓX of nearly uniform smoothness of a reflexive Banach space satisfies , then every bounded closed convex subset of X has the fixed point property for nonexpansive mappings. In particular, uniformly nonsquare Banach spaces have this property since they are properly included in this class of spaces. This answers a long-standing question in the theory.