## 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

### Keywords:

distances; critical set; regular unit simplex
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### References:

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