{
  "abstract": "Let $Ω$ be a bounded convex domain in the plane and let $R$ be a minimum-area rectangle containing $\\overlineΩ$. We prove that if the ratio of the shorter side of $R$ to its longer side is at most $ρ\\approx 0.443485$, then the first nonzero Neumann eigenvalue of $Ω$ is bounded from below by that of $R$. Moreover, equality holds if and only if $Ω$ coincides with the interior of $R$. The proof is based on a flux identity for the means of a test function over vertical sections of $Ω$, combined with weighted estimates involving the distributional second derivatives of the boundary functions. Finally, we show that the threshold $ρ\\approx0.443485$ is nearly optimal by constructing, for every ratio in $(0.457994,1/2)$, trapezoids that are arbitrarily close to their minimum-area enclosing rectangle and for which the inequality fails.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23439",
  "author_total": 3,
  "authors": [
    {
      "name": "Xiaolong Li"
    },
    {
      "name": "Jin Sun"
    },
    {
      "name": "Kui Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/565fcpc5c906y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23439",
  "fulltext_url": null,
  "id": "565fcpc5c906y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23439",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7222953256",
  "pdf_url": "https://arxiv.org/pdf/2610.23439",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/565fcpc5c906y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.23439",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23439"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Minimum-Area Enclosing Rectangles and the First Neumann Eigenvalue of Planar Convex Domains",
  "title_text": "Minimum-Area Enclosing Rectangles and the First Neumann Eigenvalue of Planar Convex Domains",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/565fcpc5c906y",
  "venue": "arXiv",
  "versions": []
}