Abstract
A theorem of van der Waerden 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 \(n\geqslant 2,\) 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.
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Alexandrov, V. An analogue of a theorem of van der Waerden, and its application to two-distance preserving mappings. Period Math Hung 72, 252–257 (2016). https://doi.org/10.1007/s10998-016-0136-1
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DOI: https://doi.org/10.1007/s10998-016-0136-1