An analogue of a van der Waerden's theorem and its application to two-distance preserving mappings

Abstract

The van der Waerden's theorem reads that an equilateral pentagon in Euclidean 3-space ${\mathbb{E}}^3$ with all diagonals of the same length is necessarily planar and its vertex set coincides with the vertex set of some convex regular pentagon. We prove the following many-dimensional analogue of this theorem: for $n⩾2$, every $n$-dimensional cross-polytope in ${\mathbb{E}}^{2n-2}$ with all diagonals of the same length and all edges of the same length necessarily lies in ${\mathbb{E}}^n$ and hence is a convex regular cross-polytope. We also apply our theorem to the study of two-distance preserving mappings of Euclidean spaces.

6 pages; in version 2, a typo is corrected in the abstract

Cited by in corpus (1)

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