On equilateral simplices in normed spaces
Peter Braß · 1997
: 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 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. 1. Introduction The question whether each d-dimensional normed space contains d+ 1 points at pairwise distance one, i.e. an equilateral simplex, was first raised by Petty in 1971 [6]. This seems obvious at first, especially in the equivalent packing version: each convex body K admits a packing (K + t i ) d+1 i=1 of d + 1 pairwise touching translates. But it turned out much more difficult, as illustrated by the following near-counterexample constructed by Petty: define a norm on IR d by fl fl fl(x 1 ; : : : ; x d ) fl fl fl : = jx 1 j + q x 2 2 + \\Delta \\Delta \\Delta + x 2 d...