On the orthogonal Grünbaum partition problem in dimension three

TL;DR

The paper shows that not every mass in $\mathbb{R}^d$ has an orthogonal equipartition by $d$ hyperplanes for $d\ge 3$, using a mass supported on the moment curve to establish nonexistence. It also provides an explicit $O(n^7)$ algorithm to detect orthogonal equipartitions for $8n$ points in $\mathbb{R}^3$ and reports that random $8$-point sets almost always admit such a partition, suggesting a high practical likelihood despite the theoretical obstruction. By connecting topological intuition with concrete geometric constructions and computation, the work informs the orthogonal variant of the Grünbaum-Hadwiger-Ramos mass-partition framework and offers avenues for improved bounds and algorithms in three dimensions.

Abstract

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

Theorems & Definitions (8)

• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof
• proof : Proof of Theorem \ref{['thm:measure']}
• Lemma 6