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For every normed space Z, we note its closed unit ball and unit sphere by Bz and Sz, respectively. Let X and Y be normed spaces such that Sx is Lipschitz homeomorphic to SXeR, and Sy is Lipschitz homeomorphic to SYeR. We prove that the following are equivalent: 1. X is Lipschitz homeomorphic to Y. 2. BX is Lipschitz homeomorphic to By. 3. SX is Lipschitz homeomorphic to Sy. This result holds also in the uniform category, except (2 or 3) => 1 which is known to be false.

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