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IN ) AND LIPSCHITZ ISOMORPHISMS

We show that the class of subspaces of c 0 (IN) is stable under Lipschitz iso-morphisms. The main corollary is that any Banach space which is Lipschitz-isomorphic to c 0 (IN) is linearly isomorphic to c 0 (IN). The proof relies in part on an isomorphic characterization of subspaces of c 0 (IN) as separable spaces having an equivalent norm such that the weak-star and norm topologies quantitatively agree on the dual unit sphere. Estimates on the Banach-Mazur distances are provided when the Lipschitz constants of the isomorphisms are small. The quite different non separable theory is also investigated.