Periodica Mathematica Hungarica

, Volume 72, Issue 2, pp 252–257 | Cite as

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

  • Victor AlexandrovEmail author


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.


Euclidean space Pentagon Cross-polytope Cayley–Menger determinant Beckman–Quarles theorem 

Mathematics Subject Classification

52B11 52B70 52C25 51K05 


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2016

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Department of PhysicsNovosibirsk State UniversityNovosibirskRussia

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