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We show that any total preorder on a set with $\binom{n}{2}$ elementscoincides with the order on pairwise distances of some point collection of size$n$ in $\mathbb{R}^{n-1}$. For linear orders, a collection of $n$ points in$\mathbb{R}^{n-2}$ suffices. These bounds turn out to be optimal. We also findan optimal bound in a bipartite version for total preorders and a near-optimalbound for a bipartite version for linear orders. Our arguments include toolsfrom convexity and positive semidefinite quadratic forms.

On prescribing total preorders and linear orders to pairwise distances of points in Euclidean space