{
  "abstract": "The Blaschke--Lebesgue problem asks for convex bodies of minimum volume among all convex bodies of prescribed constant width. In the plane, the minimizer is the Reuleaux triangle, whereas the corresponding three-dimensional problem remains open and is also known as Meissner's conjecture. In this paper, we establish the lower bound $(4π/33)d^3 \\simeq 0.380799\\,d^3$ for the volume of any three-dimensional convex body of constant width d. This improves upon Chakerian's lower bound, approximately $0.364916\\,d^3$, although it remains below the volume of the conjectured minimizers, Meissner's tetrahedra, whose volume is approximately $0.419860\\,d^3$. The proof is based on a support-function formulation, spectral estimates via spherical harmonics, and Bochner's formula. We also show that the resulting lower bound can be interpreted as a Lagrange dual bound for the associated concave quadratic minimization problem. This dual viewpoint suggests possible routes toward sharper lower bounds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: Journal of Mathematical Analysis and Applications, vol. 566(1), no. 131038 (2027)",
  "arxiv": "2606.01754",
  "author_total": 1,
  "authors": [
    {
      "name": "Akatsuki Nishioka"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2feac6ahw563k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.01754",
  "fulltext_url": null,
  "id": "2feac6ahw563k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.01754",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.MG"
    ],
    [
      "arxiv",
      "math.OC"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2606.01754",
  "pmcid": null,
  "pmid": null,
  "published": "2026-06-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2feac6ahw563k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2606.01754",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.01754"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "An Improved Lower Bound for the Three-Dimensional Blaschke--Lebesgue Problem from Spectral and Dual Perspectives",
  "title_text": "An Improved Lower Bound for the Three-Dimensional Blaschke--Lebesgue Problem from Spectral and Dual Perspectives",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/2feac6ahw563k",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.1016/j.jmaa.2026.131038",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}