Type: Article

On the orthogonal Grünbaum partition problem in dimension three

Maldonado, Gerardo L.; Roldán-Pensado, Edgardo;

Abstract:

“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.”