{
  "abstract": "We prove, with sum-of-squares certificates checked in exact arithmetic, that every cyclic $4$-polytope with seven vertices inscribed in $\\mathbb{S}^{3}$ has volume at most $2/5$. Hence no $4$-polytope of maximal volume with seven vertices inscribed in $\\mathbb{S}^{3}$ is cyclic, answering a question of Horváth and Lángi for $d = 4$, and with their theorem the maximal volume is $\\sqrt{3}/4$. With eight vertices, the same method bounds the volume of every simplicial type other than the cross-polytope by $13/20$, so the regular cross-polytope, of volume $2/3$, is up to orthogonal transformations the unique maximiser.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26886",
  "author_total": 1,
  "authors": [
    {
      "name": "Csaba Medgyes"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/22gff7rzts7k3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26886",
  "fulltext_url": null,
  "id": "22gff7rzts7k3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26886",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7223157074",
  "pdf_url": "https://arxiv.org/pdf/2610.26886",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/22gff7rzts7k3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.26886",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26886"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Inscribed 4-polytopes of maximal volume with seven and eight vertices",
  "title_text": "Inscribed 4-polytopes of maximal volume with seven and eight vertices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/22gff7rzts7k3",
  "venue": "arXiv",
  "versions": []
}