This is an announcement for the paper "A lower bound for the equilateral number of normed spaces" by Konrad J Swanepoel and Rafael Villa.
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).
Archive classification: Metric Geometry; Functional Analysis
Mathematics Subject Classification: 46B04 (Primary); 46B20, 52A21, 52C17 (Secondary)
Remarks: 5 pages
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Submitted from: swanekj@unisa.ac.za
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