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arXiv · 2304.05535

Total orders realizable as the distances between two sets of points

Gerardo L. Maldonado·Miguel Raggi Pérez·Edgardo Roldán-Pensado
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Abstract

In this note we give a negative answer to a question proposed by Almendra-Hern\'andez and Mart\'inez-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.

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Gerardo L. Maldonado, Miguel Raggi Pérez, Edgardo Roldán-Pensado. 2023-04-11. Total orders realizable as the distances between two sets of points. https://arxiv.org/abs/2304.05535

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