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An analogue of a theorem of van der Waerden, and its application to two-distance preserving mappings. (English) Zbl 1389.52014
A famous theorem of B. L. van der Waerden [Elem. Math. 25, 73–78 (1970; Zbl 0196.24101)] says that a regular pentagon in \(3\)-space (i.e., being equilateral plus having diagonals of equal length) is necessarily planar and a convex regular pentagon. Inspired by this, the author proves the following nice analogous result: For any \(n > 1\), every equilateral \(n\)-dimensional cross-polytope embedded into \((2n-2)\)-space with all diagonals of the same length, lies in \(n\)-space and is therefore a convex regular cross-polytope. The author applies this also to two-distance preserving mappings (in the spirit of the Beckman-Quarles theorem).

52B11 \(n\)-dimensional polytopes
52B70 Polyhedral manifolds
52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
51K05 General theory of distance geometry
Full Text: DOI
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