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

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.