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arXiv · 2404.01504

On the orthogonal Gr\"unbaum partition problem in dimension three

Gerardo L. Maldonado·Edgardo Roldán-Pensado
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Abstract

Gr\"unbaum's equipartition problem asked if for any measure $\mu$ on $\mathbb{R}^d$ there are always $d$ hyperplanes which divide $\mathbb{R}^d$ into $2^d$ $\mu$-equal parts. This problem is known to have a positive answer for $d\le 3$ and a negative one for $d\ge 5$. A variant of this question is to require the hyperplanes to be mutually orthogonal. This variant is known to have a positive answer for $d\le 2$ and there is reason to expect it to have a negative answer for $d\ge 3$. In this note we exhibit measures that prove this. Additionally, we describe an algorithm that checks if a set of $8n$ in $\mathbb{R}^3$ can be split evenly by $3$ mutually orthogonal planes. To our surprise, it seems the probability that a random set of $8$ points chosen uniformly and independently in the unit cube does not admit such a partition is less than $0.001$.

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Gerardo L. Maldonado, Edgardo Roldán-Pensado. 2024-04-01. On the orthogonal Gr\"unbaum partition problem in dimension three. https://arxiv.org/abs/2404.01504

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