Papers › On the orthogonal Grünbaum partition problem in dimension three
On the orthogonal Grünbaum partition problem in dimension three
Gerardo L. Maldonado, Edgardo Roldán-Pensado
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Gr\"unbaum's equipartition problem asked if for any measure μ on ℝᵈ there are always d hyperplanes which divide ℝᵈ into 2ᵈ μ-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 $0.001$.
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