Spheres in infinite-dimensional normed spaces are Lipschitz contractible

Benyamini, Y.; Sternfeld, Y.

doi:10.1090/S0002-9939-1983-0699410-7

Spheres in infinite-dimensional normed spaces are Lipschitz contractible
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by Y. Benyamini and Y. Sternfeld PDF

Proc. Amer. Math. Soc. 88 (1983), 439-445 Request permission

Abstract:

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. $\inf \{ \left \| {x - Tx} \right \|:\left \| x \right \| \leqslant 1\} > 0$.

References

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