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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).

MSC:
52B11 \(n\)-dimensional polytopes
52B70 Polyhedral manifolds
52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
51K05 General theory of distance geometry
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[1] Beckman, FS; Quarles, DA, On isometries of Euclidean spaces, Proc. Am. Math. Soc., 4, 810-815, (1953) · Zbl 0052.18204
[2] Bezdek, K; Connelly, R, Two-distance preserving functions from Euclidean space, Period. Math. Hung., 39, 185-200, (1999) · Zbl 0968.51009
[3] Blair, DE; Konno, T, Discrete torsion and its application for a generalized Van der waerden’s theorem, Proc. Japan Acad. A, 87, 209-214, (2011) · Zbl 1238.51006
[4] L.M. Blumenthal, Theory and Applications of Distance Geometry (Clarendon Press, Oxford, 1953) · Zbl 0050.38502
[5] Bottema, O, Pentagons with equal sides and equal angles, Geom. Dedicata, 2, 189-191, (1973) · Zbl 0272.50007
[6] H.S.M. Coxeter, Regular Polytopes (Methuen, London, 1948) · Zbl 0031.06502
[7] Dekster, BV, Nonisometric distance 1 preserving mapping \(E^2→ E^6\), Arch. Math., 45, 282-283, (1985) · Zbl 0558.57007
[8] Ch. Frayer, Ch. Schafhauser, Alpha-regular stick unknots. J. Knot Theory Ramif. 21(6), (2012). doi:10.1142/S0218216512500599 · Zbl 1239.57014
[9] Kuz’minykh, AV, Mappings preserving unit distance, Sib. Math. J., 20, 417-421, (1979) · Zbl 0427.51008
[10] O’Hara, J, The configuration space of equilateral and equiangular hexagons, Osaka J. Math., 50, 477-489, (2013) · Zbl 1271.65039
[11] Th.M. Rassias, On the Aleksandrov problem for isometric mappings. Appl. Anal. Discrete Math. 1(1), 18-28 (2007) · Zbl 1199.51004
[12] Waerden, BL, Ein satz über raümliche Fünfecke, Elem. Math., 25, 73-78, (1970) · Zbl 0196.24101
[13] Zaks, J, On mappings of \(\mathbb{Q}^d\) to \(\mathbb{Q}^d\) that preserve distances 1 and \(\sqrt{2}\) and the beckman-quarles theorem, J. Geom., 82, 195-203, (2005) · Zbl 1089.51002
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