On the orthogonal Gr\"unbaum partition problem in dimension three
<|reference_start|>On the orthogonal Gr\"unbaum partition problem in dimension three: 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 $0.001$.<|reference_end|>
@article{maldonado2024on,
    title={On the orthogonal Gr\"unbaum partition problem in dimension three},
    author={Gerardo L. Maldonado and Edgardo Rold'an-Pensado},
    journal={arXiv preprint arXiv:2404.01504},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.01504},
    primaryClass={math.CO cs.CG}
}