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 $0.001$.

翻译：格伦鲍姆等分割问题探讨的是：对于任意定义在 \(\mathbb{R}^d\) 上的测度 \(\mu\)，是否总存在 \(d\) 个超平面将 \(\mathbb{R}^d\) 划分为 \(2^d\) 个 \(\mu\)-等测部分。已知该问题在 \(d\le 3\) 时具有肯定答案，而在 \(d\ge 5\) 时具有否定答案。该问题的一个变体要求这些超平面必须彼此正交。已知这一变体在 \(d\le 2\) 时成立，且有理由推测其在 \(d\ge 3\) 时不成立。在本研究中，我们构造了证实这一推测的测度。此外，我们描述了一种算法，用于检验 \(\mathbb{R}^3\) 中的 \(8n\) 个点集能否被 \(3\) 个相互正交的平面均匀分割。令人惊讶的是，在单位立方体内均匀独立随机选取的 \(8\) 个点集无法被此类分割的概率似乎小于 \(0.001\)。