On equilateral simplices in normed spaces
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.
scientific article; zbMATH DE number 1460279