Spheres in Infinite-Dimensional Normed Spaces are Lipschitz Contractible
Y. Benyamini, Yaki Sternfeld · Proceedings of the American Mathematical Society · 1983
Let $X$ be an infinite-dimensional normed space. We prove the following: (i) The unit sphere $\{ x \in X:\left \| x \right \| = 1\}$ is Lipschitz contractible. (ii) There is a Lipschitz retraction from the unit ball of $X$ onto the unit sphere. (iii) There is a Lipschitz map $T$ of the unit ball into itself without an approximate fixed point, i.e. $\inf \{ \left \| {x - Tx} \right \|:\left \| x \right \| \leqslant 1\} > 0$.