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

TL;DR

The paper investigates which total preorders and linear orders on the set of pairwise distances among $n$ points, $D_n=inom{[n]}{2}$, can be realized as the order of Euclidean distances in $\mathbb{R}^d$. It establishes sharp dimension thresholds: any linear order on $D_n$ is realizable in $\mathbb{R}^{n-2}$, while any total preorder on $D_n$ is realizable in $\mathbb{R}^{n-1}$, with matching lower-bound constructions showing these are optimal. A bipartite variant on $B_{n,m}$ yields analogous, near-complete descriptions of the necessary dimensions for both total preorders and linear orders. The proofs leverage convexity arguments and the Schoenberg distance-matrix PSD criterion (along with Dekster–Wilker perspectives), providing both constructive upper bounds and explicit counterexamples for lower bounds. Together, these results link distance geometry to combinatorial orderings and identify precise dimensional thresholds for distance-based realizability in Euclidean spaces.

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.

Figures (6)

• Figure 1: Example of induced preorder on $D_4$
• Figure 2: Example of induced preorder on $B_{2,3}$
• Figure 3: Case n = 4
• Figure 4: Relations for order in $D_n$
• Figure 5: Proof for $n = 5$ in the plane

Theorems & Definitions (23)

• Theorem 1
• Theorem 2
• Proposition 3
• proof
• Corollary 4
• Proposition 5
• proof
• Proposition 6
• proof
• Theorem 7