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arXiv · 1504.03870

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

Victor Alexandrov
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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.

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Victor Alexandrov. 2015-04-17. An analogue of a van der Waerden's theorem and its application to two-distance preserving mappings. https://doi.org/10.1007/s10998-016-0136-1

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