{
  "abstract": "For every $1\\leq p<\\infty$ and even integer $m\\geq4$, we determine the optimal order of the $p$-moment torus inequality for $L_1$: it is $m^p n^{(1-p/2)_+}+n$, with comparison constants independent of $p,m,n$ after taking $p$-th roots. Consequently, for every $2\\leq q<\\infty$ and $1\\leq p\\leq q$, the corresponding metric cotype inequality holds at the sharp scale $m=O(n^{1/q})$. In particular, the quadratic inequality holds with $m=O(\\sqrt{n})$, answering the $L_1$ sharp metric cotype question of Mendel and Naor. The proof combines finite cut representations with a nonlinear smoothing estimate and an exact ternary rounding identity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08749",
  "author_total": 3,
  "authors": [
    {
      "name": "Qingjin Cheng"
    },
    {
      "name": "Yue Wang"
    },
    {
      "name": "Bo Xiang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0qj9yvahtwpje.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08749",
  "fulltext_url": null,
  "id": "0qj9yvahtwpje",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08749",
  "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.FA"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7211986709",
  "pdf_url": "https://arxiv.org/pdf/2609.08749",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0qj9yvahtwpje.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.08749",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08749"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Sharp Metric Cotype Inequalities for $L_1$ via Nonlinear Cut Smoothing",
  "title_text": "Sharp Metric Cotype Inequalities for L_1 via Nonlinear Cut Smoothing",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/0qj9yvahtwpje",
  "venue": "arXiv",
  "versions": []
}