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

作者：Alexandrov Victor^*

来源：Periodica Mathematica Hungarica, 2016, 72(2): 252-257.

DOI：10.1007/s10998-016-0136-1

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

A theorem of van der Waerden reads that an equilateral pentagon in Euclidean 3-space 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 E2n-2 with all diagonals of the same length and all edges of the same length necessarily lies in En and hence is a convex regular cross-polytope. We also apply our theorem to the study of two-distance preserving mappings of Euclidean spaces.

• 出版日期2016-6