## On prescribing total orders and preorders to pairwise distances of points in Euclidean space.(English)Zbl 1487.52010

Summary: We show that any total preorder on a set of the form $$\begin{pmatrix} X \\ 2 \end{pmatrix}$$ where $$X$$ has $$n$$ elements coincides with the order on pairwise distances of some point collection of size $$n$$ in $$\mathbb{R}^{n-1}$$. For total orders, a collection of $$n$$ points in $$\mathbb{R}^{n-2}$$ suffices. These bounds turn out to be optimal. We also find an optimal bound in a bipartite version for total preorders and a near-optimal bound for a bipartite version for total orders. Our arguments include tools from convexity and positive semidefinite quadratic forms.

### MSC:

 52A37 Other problems of combinatorial convexity

### Keywords:

Euclidean distances; total orders; total preorders; convex sets
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### References:

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