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A lower bound for the equilateral number of normed spaces

Authors:	Konrad J Swanepoel Rafael Villa

Institution:	Department of Mathematical Sciences, University of South Africa, PO Box 392, Pretoria 0003, South Africa Rafael Villa ; Departamento Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, c/Tarfia, S/N, 41012 Sevilla, Spain

Abstract:	We show that if the Banach-Mazur distance between an -dimensional normed space and $\ell_\infty^n$ is at most , then there exist equidistant points in . By a well-known result of Alon and Milman, this implies that an arbitrary -dimensional normed space admits at least $e^{c\sqrt{\log n}}$ equidistant points, where is an absolute constant. We also show that there exist equidistant points in spaces sufficiently close to $\ell_p^n$ , $1<p<\infty$ .

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