{
  "abstract": "A polyhex is an edge-connected set of n cells of the regular hexagonal tiling, where each cell has area one. We prove that the convex hull of a polyhex has area at most (1/6)*ceiling(n^2 + 14n/3), and we show that some polyhex reaches this bound for every n. This proves a conjecture of Kurz from 2008, which asked for the weaker bound (1/6)*floor(n^2 + 14n/3 + 1). The two bounds differ exactly when 3 divides n. We checked the upper bound in the Lean 4 proof assistant with the Mathlib library. We also report a computation over all polyhexes with at most 12 cells, which shows that for these sizes only one shape reaches the maximum, up to rotation and reflection.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30310",
  "author_total": 1,
  "authors": [
    {
      "name": "Pragyaan Gaur"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6rj7chyzden8f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30310",
  "fulltext_url": null,
  "id": "6rj7chyzden8f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30310",
  "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",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214608300",
  "pdf_url": "https://arxiv.org/pdf/2609.30310",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6rj7chyzden8f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30310",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30310"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The maximum area of the convex hull of a polyhex",
  "title_text": "The maximum area of the convex hull of a polyhex",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/6rj7chyzden8f",
  "venue": "arXiv",
  "versions": []
}