{
  "abstract": "We study arc length on the first-quadrant unit curve $x^p+y^p=1$, $p>1$, when length is measured with respect to an $L^r$ norm. Denoting this length by $L(p,r)$, we prove the Hölder-duality identity $L(p,r)=L(r^*,p^*)$, where $p^*=p/(p-1)$ and $r^*=r/(r-1)$. The identity is interpreted as a special case of the symmetry of mixed area (equivalently, of the anisotropic perimeter pairing). For the distinguished dual metric $r=p^*$, the length element agrees with the areal angular element: $L(p,p^*)=\\int_γ(x\\,dy-y\\,dx)=\\frac1p B(1/p,1/p)$. Thus the familiar coincidence between arc length and angular parameter on the Euclidean unit circle is the self-dual case $p=2$ of a more general relation between $L^p$ geometry and its dual metric. As a consequence, the Euclidean arc length of the $L^p$ unit curve admits a dual trigonometric representation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03693",
  "author_total": 1,
  "authors": [
    {
      "name": "Jacinto Joaquín Arichabala Concha"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/egta1r7gtcaw9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03693",
  "fulltext_url": null,
  "id": "egta1r7gtcaw9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03693",
  "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.GM"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03693",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/egta1r7gtcaw9.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.03693",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03693"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "Hölder Duality and Self-Dual Arc Length on $L^p$ Unit Curves",
  "title_text": "Hölder Duality and Self-Dual Arc Length on L^p Unit Curves",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/egta1r7gtcaw9",
  "venue": "arXiv",
  "versions": []
}