{
  "abstract": "Let $δ>0$ be fixed, let $m = \\lceil(1+δ)n\\rceil$, and let $Γ$ be an $n\\times m$ matrix with independent standard Gaussian entries. For the Gaussian Gluskin polytope $G_m = Γ(B_1^m)$ we prove $$ \\p\\left\\{d_{\\mathrm{BM}}(G_m,B_1^n) \\ge c_δn^{5/8}(\\log n)^{-1/8}\\right\\} \\ge 1-Ce^{-cn}. $$ Consequently, for all sufficiently large $n$, $$ R_1(n)\\ge c n^{5/8}(\\log n)^{-1/8}. $$ The proof uses an exact coefficient quantization, a simultaneous Gaussian compression in coefficient quotients, and an inverse-trace estimate for arbitrary inscribed bases.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.17743",
  "author_total": 1,
  "authors": [
    {
      "name": "Omer Friedland"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/342ntztca301s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.17743",
  "fulltext_url": null,
  "id": "342ntztca301s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.17743",
  "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.PR"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7204216709",
  "pdf_url": "https://arxiv.org/pdf/2608.17743",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/342ntztca301s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2608.17743",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.17743"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A $5/8$ Lower Bound on the Banach--Mazur Distance to the Cross-Polytope",
  "title_text": "A 5/8 Lower Bound on the Banach--Mazur Distance to the Cross-Polytope",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/342ntztca301s",
  "venue": "arXiv",
  "versions": []
}