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 Lipschitz homeomorphic to $S_{X \oplus R}$, and $S_Y$ is Lipschitz homeomorphic to $S_{Y \oplus R}$.

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) $\Rightarrow$ 1 which is known to be false.