{
  "abstract": "Posed by Lebesgue in 1914, the universal covering problem asks for the smallest-area planar convex set containing a congruent copy of every set of diameter at most one. We introduce an exact Reuleaux-type variational hierarchy for this constant: its monotone finite-arc values $Λ_M$ satisfy $a_{\\mathrm{Leb}}=\\lim_{M\\to\\infty}Λ_M$, and each level is a continuous finite-dimensional problem. We prove $0\\le a_{\\mathrm{Leb}}-Λ_M\\le C M^{-2}$, giving a controlled finite-arc route to the constant itself. As a certified low-order realization, an outward-rounded interval certificate for a regular finite Reuleaux subtest proves $a_{\\mathrm{Leb}}\\ge0.834$, improving the lower-bound benchmark established by Brass and Sharifi in 2005.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01284",
  "author_total": 1,
  "authors": [
    {
      "name": "Shuai Zeng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4r708hp36bn3z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01284",
  "fulltext_url": null,
  "id": "4r708hp36bn3z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01284",
  "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"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7206160608",
  "pdf_url": "https://arxiv.org/pdf/2609.01284",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4r708hp36bn3z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.01284",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01284"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "An exact hierarchy for Lebesgue's universal covering constant and a certified 0.834 lower bound",
  "title_text": "An exact hierarchy for Lebesgue's universal covering constant and a certified 0.834 lower bound",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/4r708hp36bn3z",
  "venue": "arXiv",
  "versions": []
}