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

TL;DR

This paper characterizes the conditions under which total preorders and linear orders on pairwise distances can be realized by point configurations in Euclidean space, establishing optimal bounds for such realizations.

Contribution

It provides the first precise bounds on the minimal dimension needed to realize any total preorder or linear order of pairwise distances in Euclidean space, with proofs of optimality.

Findings

Any total preorder on inom{n}{2} elements can be realized in \u211d^{n-1}.

Any linear order on inom{n}{2} elements can be realized in ^{n-2}.

Bounds are proven to be optimal for both total preorders and linear orders.

Abstract

We show that any total preorder on a set with $\binom{n}{2}$ elements coincides 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 find an optimal bound in a bipartite version for total preorders and a near-optimal bound for a bipartite version for linear orders. Our arguments include tools from convexity and positive semidefinite quadratic forms.