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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A lower bound for the equilateral number of normed spaces


Authors: Konrad J. Swanepoel and Rafael Villa
Journal: Proc. Amer. Math. Soc. 136 (2008), 127-131
MSC (2000): Primary 46B04; Secondary 46B20, 52A21, 52C17
Published electronically: August 30, 2007
MathSciNet review: 2350397
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Abstract: We show that if the Banach-Mazur distance between an $ n$-dimensional normed space $ X$ and $ \ell_\infty^n$ is at most $ 3/2$, then there exist $ n+1$ equidistant points in $ X$. By a well-known result of Alon and Milman, this implies that an arbitrary $ n$-dimensional normed space admits at least $ e^{c\sqrt{\log n}}$ equidistant points, where $ c>0$ is an absolute constant. We also show that there exist $ n$ equidistant points in spaces sufficiently close to $ \ell_p^n$, $ 1<p<\infty$.


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Additional Information

Konrad J. Swanepoel
Affiliation: Department of Mathematical Sciences, University of South Africa, PO Box 392, Pretoria 0003, South Africa

Rafael Villa
Affiliation: Departamento Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, c/Tarfia, S/N, 41012 Sevilla, Spain
Email: villa@us.es

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08916-2
PII: S 0002-9939(07)08916-2
Received by editor(s): March 23, 2006
Received by editor(s) in revised form: September 1, 2006
Published electronically: August 30, 2007
Additional Notes: This material is based upon work supported by the South African National Research Foundation under Grant number 2053752. The second author thanks the DGES grant BFM2003-01297 for financial support. Parts of this paper were written during a visit of the second author to the Department of Mathematical Sciences, University of South Africa, in January 2006.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.