تقرير
On the orthogonal Gr\'unbaum partition problem in dimension three
| العنوان: | On the orthogonal Gr\'unbaum partition problem in dimension three |
|---|---|
| المؤلفون: | Maldonado, Gerardo L., Roldán-Pensado, Edgardo |
| سنة النشر: | 2024 |
| المجموعة: | Computer Science Mathematics |
| مصطلحات موضوعية: | Mathematics - Combinatorics, Computer Science - Computational Geometry |
| الوصف: | Gr\"unbaum's equipartition problem asked if for any measure 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 $0.001$. |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2404.01504 |
| رقم الأكسشن: | edsarx.2404.01504 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |