{
  "abstract": "We prove that round circles $C_L$ are the unique minimisers of integral Menger curvature among embedded closed $C^1$ curves $γ$ of prescribed length $L$ for every $p\\geq 2$, i.e. $M_p(C_L) \\leq M_p(γ)$, with equality if and only if $γ$ parametrises a round circle. The proof combines a three-point symmetrisation, an analysis of diagonal truncations, and the periodic Wirtinger inequality. For $p>3$, known self-avoidance and regularity results extend the minimisation result to closed rectifiable curves without assuming embeddedness or smoothness, provided the length of the curve image equals $L$. In particular, $M_p$ detects the unknot and is basic for $p>3$, thereby answering a question posed repeatedly in the literature.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15981",
  "author_total": 2,
  "authors": [
    {
      "name": "Alexander Dohmen"
    },
    {
      "name": "Sebastian Scholtes"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2nbvkh87vypw8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15981",
  "fulltext_url": null,
  "id": "2nbvkh87vypw8",
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  "landing_url": "https://arxiv.org/abs/2610.15981",
  "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.CA"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
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  "openalex": "W7221806949",
  "pdf_url": "https://arxiv.org/pdf/2610.15981",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2nbvkh87vypw8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.15981",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15981"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A Sharp Lower Bound for Integral Menger Curvature",
  "title_text": "A Sharp Lower Bound for Integral Menger Curvature",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/2nbvkh87vypw8",
  "venue": "arXiv",
  "versions": []
}