On the Lipschitz classification of normed spaces, unit balls, and spheres

For every normed space Z, we note its closed unit ball and unit sphere by B -Z and S-Z, respectively. Let X and Y be normed spaces such that S-X is Lip schitz homeomorphic to S-X (+) (R), and S-Y is Lipschitz homeomorphic to S- Y (+) (R). We prove that the following are equivalent: 1. X is Lipschitz homeomorphic to Y. 2. B-X is Lipschitz homeomorphic to B-Y. 3. S-X is Lipschitz homeomorphic to S-Y. This result holds also in the uniform category, except (2 or 3) double righ t arrow 1 which is known to be false.