On the orthogonal Grünbaum partition problem in dimension three

Abstract

Grünbaum's equipartition problem asked if for any measure $\hat{I}¼$ on ${\mathbb{R}}^d$ there are always $d$ hyperplanes which divide ${\mathbb{R}}^d$ into $2^d$ $\hat{I}¼$-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 $0.001$.