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A lower bound for the equilateral number of normed spaces

Konrad J Swanepoel·Rafael Villa
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Abstract

We show that if the Banach-Mazur distance between an n-dimensional normed space X and ell infinity 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 n-dimensional ell p (1 < p < infinity).

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Konrad J Swanepoel, Rafael Villa. 2006-03-27. A lower bound for the equilateral number of normed spaces. https://doi.org/10.1090/s0002-9939-07-08916-2

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