{
  "abstract": "Let $B_1^n$ be the standard cross-polytope in $\\mathbb R^n$, let $g_1,\\ldots,g_m$ be independent standard Gaussian vectors in $\\mathbb R^n$, and set $G_m=\\operatorname{conv}{\\pm g_1,\\ldots,\\pm g_m}$. For $m=n^3$ it is proved that $$ \\mathbb P\\left\\{d_{\\mathrm{BM}}(G_m,B_1^n)\\geqslant c n^{5/8}(\\ln n)^{-5/8}\\right\\}\\geqslant 1-\\frac2n $$ for a suitable absolute constant $c>0$. This independently improves the polynomial exponent $4/7$ in Friedland's preceding work. Independent concurrent work of Friedland, which appeared after completion of the present manuscript, obtains the same polynomial exponent with the stronger logarithmic factor $(\\ln n)^{-1/4}$ by a different argument. The proof uses Friedland's discretization and conditioning argument together with the $K/U$ decomposition. A selected family of $K$ vectors is suppressed and the remaining $K$ vectors are quotiented out. In the resulting quotient simultaneous bounds are proved for every top-dimensional exterior product formed from the suppressed $K$ vectors and the $U$ vectors. A Dvoretzky-Rogers selection after L\"owner normalization converts these determinant estimates into a bound for the minimum volume ellipsoid of the whole projected polytope and Maurey's empirical method then gives the required Gaussian measure estimate.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27796",
  "author_total": 1,
  "authors": [
    {
      "name": "Antonios Hmadi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/crjy305ydkhr1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27796",
  "fulltext_url": null,
  "id": "crjy305ydkhr1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27796",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/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": "W7214218850",
  "pdf_url": "https://arxiv.org/pdf/2609.27796",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/crjy305ydkhr1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.27796",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27796"
    }
  ],
  "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": "Gaussian polytopes with large Banach-Mazur distance to the cross-polytope",
  "title_text": "Gaussian polytopes with large Banach-Mazur distance to the cross-polytope",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/crjy305ydkhr1",
  "venue": "arXiv",
  "versions": []
}