{
  "abstract": "In this article, we prove the sharp form of the fractional isoperimetric inequality in the plane, originally posed by Maz'ya [Problem 1, Integral Equations Operator Theory, 2018]. More precisely, we show that, for any given $s\\in(0,1)$ and any bounded domain $Ω\\subset \\mathbb{R}^2$ with $C^1$ boundary, $$ P_s(Ω) \\le \\frac{π^{s-\\frac{1}{2}}Γ\\left(\\frac{3-s}{2}\\right)}{s(1-s)Γ\\left(\\frac{4-s}{2}\\right)}\\left[\\mathcal{H}^1(\\partialΩ)\\right]^{2-s}, $$ where $P_s$ denotes the fractional $s$-perimeter, $Γ$ denotes the Gamma function, and $\\mathcal{H}^{1}$ denotes the $1$-dimensional Hausdorff measure on $\\mathbb{R}^2$. The constant is sharp and equality is attained by the disk. The proof proceeds in three steps: reducing the fractional perimeter to a chord functional, reducing connected domains to convex bodies, and proving the sharp convex chord inequality via a fractional Willmore-type inequality, a variational formula along outer parallel bodies, and asymptotic analysis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19052",
  "author_total": 5,
  "authors": [
    {
      "name": "Xiaosheng Lin"
    },
    {
      "name": "Dachun Yang"
    },
    {
      "name": "Sibei Yang"
    },
    {
      "name": "Wen Yuan"
    },
    {
      "name": "Yangyang Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c5bxrrmg6yj26.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19052",
  "fulltext_url": null,
  "id": "c5bxrrmg6yj26",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19052",
  "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.AP"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7213527824",
  "pdf_url": "https://arxiv.org/pdf/2609.19052",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c5bxrrmg6yj26.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2609.19052",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19052"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Sharp Planar Fractional Isoperimetric Inequality",
  "title_text": "A Sharp Planar Fractional Isoperimetric Inequality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/c5bxrrmg6yj26",
  "venue": "arXiv",
  "versions": []
}