Edge lengths guaranteed to form a simplex. (English) Zbl 0608.52007

Obviously there exists a minimum number \(\lambda =\lambda (n)\in [0,1]\) such that points \(p_ 1,p_ 2,...,p_{n+1}\) with prescribed mutual distances \(\ell _{ij}=d(p_ i,p_ j)\) exist in \({\mathbb{R}}^ n\) provided \(\lambda \leq \ell _{ij}\leq 1\), \(i\neq j\). A matrix \((\ell _{ij})\) with \(\ell _{ii}=0\), \(\ell _{ij}=\ell _{ji}\geq 0\) is called allowable. An allowable matrix of order \(n+1\) is called realizable if there exist points \(p_ 1,p_ 2,...,p_{n+1}\in {\mathbb{R}}^ n\) (a realization), satisfying \(d(p_ i,p_ j)=\ell _{ij}\). A set \(p_ 1,p_ 2,...,p_{n+1}\in {\mathbb{R}}^ n\) is called critical if the matrix \(L=(\ell _{ij})\) of its mutual distances satisfies \((i)\quad \lambda (n)\leq \ell _{ij}\leq 1;\quad \min _{i\neq j}\ell _{ij}=\lambda (n).\) (ii) In any neighbourhood of the matrix L there exists an allowable but nonrealizable matrix.
Theorem. In each critical set of \(n+1\) points, \(n\geq 2\), \([(n+1)/2]\) of them are vertices of a regular unit simplex. The rest of the points are vertices of another regular unit simplex. The two simplexes have a common center and their flats are orthogonal. The distance between vertices from distinct simplexes is \(\lambda\) (n). Thus \(\lambda (n)=\sqrt{1- 2(n+1)/n(n+2)}\) for even \(n\geq 2\) and \(\lambda (n)=\sqrt{1-2/(n+1)}\) for odd \(n\geq 3.\)
Remark. The dimensions of the two simplexes are \([(n+1)/2]-1\) and \(n- [(n+1)/2].\) Thus the critical set determines a hyperplane in \({\mathbb{R}}^ n\) and is the set of vertices of a degenerate simplex in \({\mathbb{R}}^ n\).


52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
52Bxx Polytopes and polyhedra
52A40 Inequalities and extremum problems involving convexity in convex geometry
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[1] F. S. Beckman andD. A. Quarles, On isometries of Euclidean spaces. Proc. Amer. Math. Soc.4, 810-815 (1953). · Zbl 0052.18204 · doi:10.1090/S0002-9939-1953-0058193-5
[2] D. M.Bloom, Linear Algebra and Geometry. Cambridge University Press 1979. · Zbl 0405.15001
[3] L. M.Blumenthal, Theory and Applications of Distance Geometry. Oxford 1953.
[4] L. M. Blumenthal, A budget of curiosa metrica. Amer. Math. Monthly66, 453-460 (1959). · Zbl 0092.39401 · doi:10.2307/2310624
[5] O. Bottema, Pentagons with equal sides and equal angles. Geom. Dedicata2, 189-191 (1973). · Zbl 0272.50007 · doi:10.1007/BF00147855
[6] B. V. Dekster, Non-isometric distance 1 preserving mappingE 2?E 6. Arch. Math.45, 282-283 (1985). · Zbl 0558.57007 · doi:10.1007/BF01275582
[7] B. V.Dekster and J. B.Wilker, Finite geometries as real configurations; in: 3. Kolloquium ?ber diskrete Geometrie, Institut f?r Mathematik der Universit?t Salzburg 1985. · Zbl 0567.51011
[8] P. M.Gruber and R.Schneider, Problems in Geometric Convexity. Proc. Geom. Sympos., Siegen 1978, Basel 1979.
[9] B.Gr?nbaum, Polygons; in: The Geometry of Metric and Linear Spaces. Berlin-Heidelberg-New York 1975.
[10] F. Herzog, Completely tetrahedral Simplexes. Amer. Math. Monthly66, 460-464 (1959). · Zbl 0092.39402 · doi:10.2307/2310625
[11] C. L. Morgan, Embedding metric spaces in Euclidean space. J. Geometry5, 101-107 (1974). · Zbl 0288.52012 · doi:10.1007/BF01954540
[12] A. Neumaier, Distance matrices, dimension, and conference graphs. Indag. Math.43, 385-391 (1981). · Zbl 0523.05019
[13] I. J. Schoenberg, Remarks to Maurice Fr?chet’s article Sur la d?finition axiomatique d’une classe d’espace distanci?s vectoriellement applicable sur l’espace de Hilbert. Ann. of Math.36, 724-732 (1935). · doi:10.2307/1968654
[14] I. J. Schoenberg, Linkages and distance geometry, I. Linkages, II. On sets ofn+2 points inE n that are most nearly equilateral. Indag. Math.31, 43-63 (1969). · Zbl 0169.24702
[15] J. J. Seidel, Quasiregular two-distance sets. Indag. Math.31, 64-70 (1969). · Zbl 0167.50801
[16] B. L. van der Waerden, Ein Satz ?ber R?umliche F?nfecke. Elem. Math.25, 73-78 (1970). · Zbl 0196.24101
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