{
  "abstract": "We prove that every convex body $K\\subset\\mathbb{R}^3$ of constant width $w>0$ satisfies \\[ \\operatorname{Vol}(K)\\ge \\frac{1651}{4000}w^3=0.41275\\,w^3. \\] This improves the certified lower bound $0.411040473721188\\ldots\\,w^3$ announced by Hyra and the earlier spectral bound $4πw^3/33$ of Nishioka. The main new ingredient is a contact-tetrahedron estimate near equality in Jung's circumradius inequality. Four balanced contacts of a minimum circumscribed ball form a near-regular tetrahedron; constant width then forces, beyond the tetrahedron itself, spherical regions over its faces, spindle sectors along its edges, and additional face-edge sectors. These regions can be selected with pairwise disjoint interiors and quantified by the total squared-edge deficit. A sharp convexity argument relates that deficit to the circumradius, and a symmetrization argument controls the four face contributions with only quadratic loss. This gives the analytic bound $\\operatorname{Vol}(K)>0.41275318$ near the Jung endpoint. For the complementary circumradius range we use Hyra's radial transport/tangent-majorant mechanism together with closed-form contact and antipodal profiles. The antipodal contribution has an elementary primitive. For the contact contribution, a second-order concavity estimate converts the angular Jacobian into an explicit rational lower density; that density is monotone in the circumradius. The whole intermediate range therefore reduces to a single endpoint calculation, giving $\\operatorname{Vol}(K)>0.41275168$. For smaller circumradius the tangent-majorant inequality with its nonnegative gap discarded already gives $\\operatorname{Vol}(K)>0.41308109$. The proof is entirely analytic and requires no computer-assisted certificate.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13964",
  "author_total": 1,
  "authors": [
    {
      "name": "Yunjiang Jiang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/54b7mwab38c25.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13964",
  "fulltext_url": null,
  "id": "54b7mwab38c25",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13964",
  "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",
      "cs.CG"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7221804592",
  "pdf_url": "https://arxiv.org/pdf/2610.13964",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/54b7mwab38c25.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.13964",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13964"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A 0.41275 lower bound for the three-dimensional Blaschke-Lebesgue problem",
  "title_text": "A 0.41275 lower bound for the three-dimensional Blaschke-Lebesgue problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/54b7mwab38c25",
  "venue": "arXiv",
  "versions": []
}