A lower bound for the equilateral number of normed spaces

Abstract

We show that if the Banach-Mazur distance between an n n -dimensional normed space X X and ℓ ∞ n \ell _\infty ^n is at most 3 / 2 3/2 , then there exist n + 1 n+1 equidistant points in X X . By a well-known result of Alon and Milman, this implies that an arbitrary n n -dimensional normed space admits at least e c log ⁡ n e^{c\sqrt {\log n}} equidistant points, where c > 0 c>0 is an absolute constant. We also show that there exist n n equidistant points in spaces sufficiently close to ℓ p n \ell _p^n , 1 > p > ∞ 1>p>\infty .