{
  "abstract": "We present a simple proof of the $\\varepsilon$-Dvoretzky conjecture, which asserts that the dependence on the approximation parameter $\\varepsilon$ in Dvoretzky's theorem is polynomial in $1/\\varepsilon$. In particular, if $n \\geq (C/\\varepsilon)^{\\ell/2+1}$, then any $n$-dimensional convex body has, through any given interior point, an $\\ell$-dimensional section that is $\\varepsilon$-close to a Euclidean ball. Here, $C > 0$ is a universal constant. We in fact obtain a sharper dependence on $\\varepsilon$. The proof is probabilistic, but uses a different probabilistic model from those employed previously. We also prove a simultaneous version of our theorem for a finite family of convex bodies with the origin in their interior, yielding a common linear subspace on which all of the bodies have nearly-Euclidean sections. Finally, we compare the radius of the approximating Euclidean ball with familiar geometric parameters, such as the mean widths of the body and its dual.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03204",
  "author_total": 2,
  "authors": [
    {
      "name": "Boaz Klartag"
    },
    {
      "name": "Shahar Moshe"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b2ng95kybzm9j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03204",
  "fulltext_url": null,
  "id": "b2ng95kybzm9j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03204",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7220419082",
  "pdf_url": "https://arxiv.org/pdf/2610.03204",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b2ng95kybzm9j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.03204",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03204"
    }
  ],
  "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": "A polynomial bound in Dvoretzky's theorem",
  "title_text": "A polynomial bound in Dvoretzky's theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/b2ng95kybzm9j",
  "venue": "arXiv",
  "versions": []
}