On equilateral simplices in normed spaces
type: Item
instance of: scholarly article
title: On equilateral simplices in normed spaces
zbMATH Open document ID: 0961.51016
published in: Beiträge zur Algebra und Geometrie
publication date: 2000-06-18T00:00:00Z
full work available at URL: https://eudml.org/doc/225134
full work available at URL: 3.html
review text: According to the abstract, the aim of this work is to improve the lower bound for the Petty's problem on the existence of equilateral simplices in normed spaces. Petty's problem asks if each \(d\)-dimensional normed space contains \(d+1\) points at pairwise distance one, i.e. an equilateral simplex [see \textit{C. M. Petty}, Proc. Am. Math. Soc. 29, 369-374 (1971; Zbl 0214.20801)]. This problem seemed obvious at first, if one looked at the equivalent packing version, but near-counterexamples constructed by Petty himself showed that the problem was much more difficult. In fact Petty showed that there are norms in \({\mathbb R}^{d}\) for which there exist nonextendable equilateral sets of four points. He also showed that each normed space of dimension at least three contains four points at pointwise distance one, conjecturing that each normed space contains \(d+1\) points at pairwise distance one (see also problem 4.1.1 of the book [\textit{A. C. Thompson}, ``Minkowski geometry'' (1996; Zbl 0868.52001)]), but no progress beyond the lower bound of four [see \textit{G. Lawlor} and \textit{F. Morgan}, Pac. J. Math. 166, No. 1, 55-83 (1994; Zbl 0830.49028) and \textit{F. Morgan}, Math. Intell. 14, No. 3, 37-44 (1992; Zbl 0765.52015)]. The author proves that for each \(k\), there exists a \(d (k\)) such that each normed space of dimension \(d\geq d (k)\) contains \(k\) points at pairwise distance one. The proof is done applying Dvoretzky's theorem [see \textit{A. Dvoretzky}, Proc. Int. Symp. Linear Spaces, 123-160 (1961; Zbl 0119.31803)] to the following theorem proved by the author in this paper: for each dimension \(d\), there is \(\varepsilon _{d}>0\) such that if \((V,\|\cdot\|)\) is a \(d\)-dimensional normed space with \((1-\varepsilon _{d})\|x\|_{\text{euclidean}}\leq \|x\|\leq 1+\varepsilon _{d})\|x\|_ {\text{euclidean}}\), for all \(x\in V\), then each equilateral set in \(V\) can be extended to an equilateral set of \(d+1\) points.
reviewed by: Ana Pereira do Vale
Mathematics Subject Classification ID: 51M20
zbMATH DE Number: 1460279
zbMATH Keywords: equilateral simplices
zbMATH Keywords: Petty's problem
zbMATH Keywords: Cayley-Menger determinant
author: Peter Brass
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1460279