{
  "abstract": "We study orthogonal $(4,3)$-equipartitions of finite measures absolutely continuous with respect to Lebesgue measure: every three of four mutually orthogonal affine hyperplanes must divide each measure into eight equal parts. For every number $m$ of measures, we determine the exact dimension threshold of Musin's polynomial criterion and express it in terms of the binary expansion of $m$. The threshold coincides with the geometric lower bound precisely when $m=2^j-1$. Consequently, $Δ^*(2^j-1,4,3)=7\\cdot2^{j-1}-2$ for every $j\\geq1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19214",
  "author_total": 3,
  "authors": [
    {
      "name": "Alexey Fakhrutdinov"
    },
    {
      "name": "Platon Khamatov"
    },
    {
      "name": "Oleg R. Musin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e1008fchnmg49.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19214",
  "fulltext_url": null,
  "id": "e1008fchnmg49",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19214",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7222239592",
  "pdf_url": "https://arxiv.org/pdf/2610.19214",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e1008fchnmg49.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19214",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19214"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Orthogonal partition by four hyperplanes",
  "title_text": "Orthogonal partition by four hyperplanes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/e1008fchnmg49",
  "venue": "arXiv",
  "versions": []
}