Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2404.01504 |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866916421116624896 |
|---|---|
| author | Maldonado, Gerardo L. Roldán-Pensado, Edgardo |
| author_facet | Maldonado, Gerardo L. Roldán-Pensado, Edgardo |
| contents | 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 $0.001$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_01504 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the orthogonal Grünbaum partition problem in dimension three Maldonado, Gerardo L. Roldán-Pensado, Edgardo Combinatorics Computational Geometry 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 $0.001$. |
| title | On the orthogonal Grünbaum partition problem in dimension three |
| topic | Combinatorics Computational Geometry |
| url | https://arxiv.org/abs/2404.01504 |