{
  "abstract": "Let $R_\\infty(n)$ be the maximal Banach--Mazur distance from an $n$-dimensional real normed space to $\\ell_\\infty^n$. We prove that $R_\\infty(n)\\ge cn^{2/3}(\\log(2n))^{-10}$. Together with the upper bound of Friedland, Naor, and Youssef, this gives $R_\\infty(n) = n^{2/3+o(1)}$. The examples are polars of symmetric Gaussian polytopes with $2n$ generators, and the lower bound holds with exponentially high probability. A joint minimization over coefficient representations and positive weights yields an exact second-variation identity and sparse coefficients. A Gaussian projection argument controls all low-rank completions of the recorded matrix. An adaptive choice of exceptional generators then reduces the remaining case to a fixed-shift Gaussian small-ball estimate over two finite nets.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36917",
  "author_total": 1,
  "authors": [
    {
      "name": "Omer Friedland"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d8c2jn1vm8rsw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36917",
  "fulltext_url": null,
  "id": "d8c2jn1vm8rsw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36917",
  "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"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7224087255",
  "pdf_url": "https://arxiv.org/pdf/2610.36917",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d8c2jn1vm8rsw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36917",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36917"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The sharp exponent for the Banach--Mazur distance to the cube",
  "title_text": "The sharp exponent for the Banach--Mazur distance to the cube",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/d8c2jn1vm8rsw",
  "venue": "arXiv",
  "versions": []
}