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


Author: Ronny Nahum
Journal: Proc. Amer. Math. Soc. 129 (2001), 1995-1999
MSC (2000): Primary 46B20
DOI: https://doi.org/10.1090/S0002-9939-00-05782-8
Published electronically: December 13, 2000
MathSciNet review: 1825907
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Abstract:

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.


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Additional Information

Ronny Nahum
Affiliation: Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel
Email: ronnyn@techunix.technion.ac.il

DOI: https://doi.org/10.1090/S0002-9939-00-05782-8
Keywords: Lipschitz, biLipschitz, uniform homeomorphism
Received by editor(s): April 20, 1998
Received by editor(s) in revised form: October 20, 1999
Published electronically: December 13, 2000
Additional Notes: This paper is a part of the author’s Ph.D. thesis, prepared at the University of Haifa under the supervision of Prof. Y. Sternfeld.
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society

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