Gr\"unbaum's equipartition problem asked if for any measure 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 $0.001$.

翻译：Grünbaum的等分问题提出：对于$\mathbb{R}^d$上的任意测度，是否存在$d$个超平面将$\mathbb{R}^d$划分为$2^d$个$\mu$-等份？已知该问题在$d\le 3$时具有肯定答案，在$d\ge 5$时则是否定答案。该问题的一个变体要求这些超平面相互正交。已知该变体在$d\le 2$时具有肯定答案，且有理由预期其在$d\ge 3$时是否定答案。本文构造了证明该结论的测度。此外，我们描述了一种算法，用于判断$\mathbb{R}^3$中的$8n$个点集能否被3个相互正交的平面均分。令人意外的是，在单位立方体中均匀独立选取的8个随机点集，其无法被如此划分的概率似乎小于0.001。