A lower bound for the equilateral number of normed spaces

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).

5 pages

References in corpus (2)

• Equilateral sets in finite-dimensional normed spaces
• A problem of Kusner on equilateral sets

Cited by in corpus (10)

• Combinatorial distance geometry in normed spaces
• Equilateral sets in uniformly smooth Banach spaces
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• Equilateral sets in infinite dimensional Banach spaces
• Equilateral Sets in Banach Spaces of th form C(K)
• Some old and new problems in combinatorial geometry I: Around Borsuk's problem
• Isometric embeddings of a class of separable metric spaces into Banach spaces
• On inscribed equilateral simplices in normed spaces
• Antipodal Hadwiger numbers of finite-dimensional Banach spaces
• A Note on Gram-Schmidt's Algorithm for a General Angle