{
  "abstract": "We study Heilbronn's triangle problem in the unit right triangle, where $n$ points are placed to maximize the smallest of the $\\binom{n}{3}$ triangle areas they span. We prove a boundary-structure result: unless all three vertices are occupied, some optimal configuration with $n \\ge 5$ has at least four points on the boundary, one edge containing two of them. With the affine $S_3$ symmetry this fixes four boundary points and $n$ orientation variables in a mixed-integer model that certifies global optimality for all $n \\le 8$ up to a relative gap of $10^{-4}$. For $n \\le 7$ this independently confirms the known exact optima, the case $n = 7$ having been proved by Zeng and Chen with symbolic computation. For $n = 8$, where only bounds were known, it narrows the gap to Cantrell's configuration fourfold in 39 minutes on one machine. The optimum there is conjectured to be the real root of a septic obtained by Chen, Zeng and Zhou, which our reconstruction confirms to 250 digits. We show its Galois group is $S_7$, so on that conjecture no expression in radicals exists.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.15021",
  "author_total": 1,
  "authors": [
    {
      "name": "Nathan Sudermann-Merx"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bmf6513q1fxk8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.15021",
  "fulltext_url": null,
  "id": "bmf6513q1fxk8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.15021",
  "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.OC"
    ],
    [
      "openalex",
      "1704|Computational Geometry and Mesh Generation"
    ]
  ],
  "openalex": "W7169509344",
  "pdf_url": "https://arxiv.org/pdf/2607.15021",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bmf6513q1fxk8.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2607.15021",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.15021"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Heilbronn's Problem in the Unit Triangle: Computational Optimality Certificates for $n \\le 8$",
  "title_text": "Heilbronn's Problem in the Unit Triangle: Computational Optimality Certificates for n ≤ 8",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:51Z",
  "url": "https://research.draconitic.com/paper/bmf6513q1fxk8",
  "venue": "arXiv",
  "versions": []
}