On fixed-point sets of nonexpansive mappings in nonstandard hulls and Banach space ultrapowers

We present a study on the structure of fixed-point sets of nonexpansive mappings in nonstandard hulls (Banach space ultrapowers), initiated by the result of Maurey (1981) about metric convexity of FixT˜. Our main motivation is a long-standing problem concerning the existence of common approximate fixed points for a family of commuting nonexpansive mappings.