{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/on-the-orthogonal-grunbaum-partition-problem","title":"On the orthogonal Grünbaum partition problem in dimension three","arxiv_id":"2404.01504","date":"2024-04-01","proceeding":null,"authors":["Gerardo L. Maldonado","Edgardo Roldán-Pensado"],"abstract":"Gr\\\"unbaum's equipartition problem asked if for any measure $\\mu$ 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 $0.001$.","url_abs":"https://arxiv.org/abs/2404.01504v3","url_pdf":"https://arxiv.org/pdf/2404.01504v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"on-the-orthogonal-grunbaum-partition-problem","repo_url":"https://github.com/xgeu2x/grunbaum-3d-partitions","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}