On Equilateral Simplices in Normed Spaces

Abstract: It is the aim of this note to improve the lower bound for the problem of Petty on the existence of equilateral simplices in normed spaces. We show that for each $k$ there is a $d(k)$ such that each normed space of dimension $d\geq d(k)$ contains $k$ points at pairwise distance one, and that if the norm is sufficiently near to the euclidean norm, the maximal equilateral sets behave like their euclidean counterparts.

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