Total orders realizable as the distances between two sets of points

TL;DR

The paper studies the problem of realizing all distance-induced orders on $[n]\times[m]$ from two point sets in $\mathbb{R}^d$, under the assumption that all cross-distances are unique. It proves a sharp lower bound by constructing a misaligned order on $[n]\times[n+1]$ that cannot be realized in $\mathbb{R}^{n-1}$, thereby showing the minimal universal dimension must satisfy $d\ge n$ (with the $n=m$ case left open). It further shows that a claimed upper bound in AM22 is flawed and provides a corrected lemma and explicit counterexamples for $d\ge4$, along with a computational exploration that supports realizability for many cases and suggests a conjecture for all permutations of $[d+1]\times[d+1]$ in $\mathbb{R}^d$. The work combines combinatorial, geometric, and computational techniques to delineate the boundary between achievable and impossible distance orders in fixed dimensions. The results motivate a more complete characterization of distance-order realizability and identify directions for future theoretical and algorithmic work.

Abstract

In this note we give a negative answer to a question proposed by Almendra-Hernández and Martínez-Sandoval. Let $n\le m$ be positive integers and let $X$ and $Y$ be sets of sizes $n$ and $m$ in $\mathbb{R}^{n-1}$ such that every pair of points in $X\cup Y$ defines a unique distance. There is a natural order on $X\times Y$ induced by the distances between the corresponding points. The question is if all possible orders on $X\times Y$ can be obtained in this way. We show that the answer is negative when $n<m$. The case $n=m$ remains open.

Figures (5)

• Figure 1: Left: The arrows represent order restrictions of Definition \ref{['def:order']}, (as a torus). Right: An example for $d=4$ of a misaligned order.
• Figure 2: Visualization of Observation \ref{['obs:dm']}. The red solid lines are the perpendicular bisectors of the blue dashed segments. Points in $P$ are blue while points from $Q$ are black.
• Figure 3: Left: Cone $C_1$, corresponding to the permutation $\sigma = (1,2,0)$, shown. Right: Cone $C_1^*$ shown.
• Figure 4: $q_0$ must be in the dashed green line and $q_{d+1}$ must be in the solid blue line, so $q_0$ and $q_{d+1}$ lie on the same side of $H$ as $O(Q_0)$.
• Figure 5: Example of the Lemma \ref{['lem:coro4']} in the plane. The green intersection must lie inside the blue simplex.

Theorems & Definitions (9)

• Definition 1
• Remark
• Theorem 2
• Lemma 4
• proof
• Lemma 5
• proof
• Proposition 6: Counterexample to Proposition 5 from AM22
• proof