{
  "abstract": "Karasev proved, in dimensions two and three, a sharp surface-area counterpart of a polyhedral extension of Vaaler's theorem [8, Theorem 1.2]. We remove the dimension restriction. More precisely, if an $n$-dimensional convex polytope $P$ contains the origin in its interior and the affine hull of every nonempty proper face of codimension $k\\in\\{1,\\ldots,n\\}$ is at distance at least $\\sqrt{k}$ from the origin, then $$ \\mathcal{H}^{n-1}(\\partial P)\\geqslant n2^n. $$ Consequently, the boundary of every $n$-dimensional linear section of the cube $[-1,1]^N$ has $(n-1)$-dimensional measure at least $n2^n$. This confirms, in all dimensions, a conjecture of Grigory M. Ivanov recorded by Karasev [8, Section 1]. The proof combines the Rogers--Karasev flag decomposition with a dimension-free Gaussian comparison for orthoschemes. Its main step is an ordered-square substitution which converts all face-distance assumptions into a pointwise domination of a single positive integral. We also disprove the naive extension to all skeletal measures, formulate a ridge-skeleton conjecture, and establish two partial results toward it.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27798",
  "author_total": 1,
  "authors": [
    {
      "name": "Michał Zwierzyński"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2whjxqz938qvj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27798",
  "fulltext_url": null,
  "id": "2whjxqz938qvj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27798",
  "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"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7214238624",
  "pdf_url": "https://arxiv.org/pdf/2609.27798",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2whjxqz938qvj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27798",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27798"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A Sharp Surface-Area Extension of Vaaler's Theorem",
  "title_text": "A Sharp Surface-Area Extension of Vaaler's Theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/2whjxqz938qvj",
  "venue": "arXiv",
  "versions": []
}