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

Gr\"unbaum's equipartition problem asked if for any measure on Rᵈ there are always d hyperplanes which divide Rᵈ into 2ᵈ -equal parts. This problem is known to have a positive answer for d 3 and a negative one for d 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 2 and there is reason to expect it to have a negative answer for d 3. In this note we exhibit measures that prove this. Additionally, we describe an algorithm that checks if a set of 8n in R³ 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.