{
  "abstract": "Kuperberg conjectured in 1983 that every convex body $K$ in the plane is contained in a quadrilateral of area at most $\\frac{3}{\\sqrt5}\\,|K|$, the extremal bodies being the affine-regular pentagons. On the basis of numerical experiments we propose a stronger conjecture of a purely polygonal nature: every convex polygon $P$ is contained in a quadrilateral of area at most $\\frac{3}{\\sqrt5}\\,|P|$ whose sides are parallel to sides or diagonals of $P$. This \\emph{chord conjecture} implies the inequality conjectured by Kuperberg. For pentagons it is a theorem of Hong, Ismailescu, Kwak and Park, of which we give a short proof by area identities. We also show that the classical bound $\\sqrt2$ remains valid for quadrilaterals with sides parallel to chords; the proof is similar to Ismailescu's proof of this bound, but it starts from an inscribed quadrilateral of maximal area instead of a circumscribed quadrilateral of minimal area. We close with a discussion of the difficulties in reaching the constant $\\frac{3}{\\sqrt5}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08168",
  "author_total": 1,
  "authors": [
    {
      "name": "Viktor Vígh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b44nfzbhkqvvy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08168",
  "fulltext_url": null,
  "id": "b44nfzbhkqvvy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08168",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7221004519",
  "pdf_url": "https://arxiv.org/pdf/2610.08168",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b44nfzbhkqvvy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.08168",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08168"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A note on Kuperberg's quadrilateral conjecture",
  "title_text": "A note on Kuperberg's quadrilateral conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/b44nfzbhkqvvy",
  "venue": "arXiv",
  "versions": []
}