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

TL;DR

This paper generalizes van der Waerden's theorem to higher dimensions, showing that certain equilateral polytopes must be regular and lie in lower-dimensional spaces, and applies this to analyze two-distance preserving mappings.

Contribution

It introduces a many-dimensional analogue of van der Waerden's theorem for cross-polytopes and explores its implications for two-distance preserving mappings in Euclidean spaces.

Findings

n-dimensional cross-polytopes with equal diagonals are regular and lie in lower-dimensional space

the theorem characterizes the structure of equilateral polytopes in higher dimensions

applications to mappings preserving two distances in Euclidean spaces

Abstract

The van der Waerden's theorem reads that an equilateral pentagon in Euclidean 3-space $\Bbb E^3$ 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 $n\geqslant 2$, every $n$-dimensional cross-polytope in $\Bbb E^{2n-2}$ with all diagonals of the same length and all edges of the same length necessarily lies in $\Bbb E^n$ and hence is a convex regular cross-polytope. We also apply our theorem to the study of two-distance preserving mappings of Euclidean spaces.