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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 \({\mathbb{R}}^ n\).

MSC:

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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