{
  "id": "2404.01504",
  "version": "v1",
  "published": "2024-04-01T22:10:15.000Z",
  "updated": "2024-04-01T22:10:15.000Z",
  "title": "On the orthogonal Grünbaum partition problem in dimension three",
  "authors": [
    "Gerardo L. Maldonado",
    "Edgardo Roldán-Pensado"
  ],
  "categories": [
    "math.CO",
    "cs.CG"
  ],
  "abstract": "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 $0.001$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-01T22:10:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "orthogonal grünbaum partition problem",
      "positive answer",
      "hyperplanes",
      "unit cube",
      "equal parts"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}