{
  "abstract": "Let $K\\subset\\R^n$ be a convex body in isotropic position. We prove the optimal mean-width estimate \\[ M^*(K)\\leq C\\sqrt{n\\log n}. \\] The main new ingredient is a geometric inequality relating the Gaussian mean of the support function to its mean under the uniform measure on $K$, obtained through a heat-flow argument. Combined with the Gaussian-log-concave comparison of Eldan and Lehec and the newly available dimension-free bound on the third-moment parameter $κ_n$, this yields the result. The boundedness of $κ_n$ also makes the mean-norm estimate of Bizeul and Klartag sharp. Combining both estimates yields \\[ M(K)M^*(K)\\leq C\\log n, \\] extending Pisier's $MM^*$ estimate to non-symmetric convex bodies and to the isotropic position.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.29458",
  "author_total": 1,
  "authors": [
    {
      "name": "Pierre Bizeul"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2e084wckcnhak.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.29458",
  "fulltext_url": null,
  "id": "2e084wckcnhak",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.29458",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7172327716",
  "pdf_url": "https://arxiv.org/pdf/2607.29458",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2e084wckcnhak.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2607.29458",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.29458"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Optimal MM* bounds for convex bodies",
  "title_text": "Optimal MM* bounds for convex bodies",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/2e084wckcnhak",
  "venue": "arXiv",
  "versions": []
}