{
  "abstract": "Let $m\\geq1$ be fixed. We consider the space $\\mathcal{P}_m(\\mathbb{R}^n)$ of real $m$-homogeneous polynomials on $\\mathbb{R}^n$, endowed with the standard Gaussian measure $γ_{m,n}$ associated with the Bombieri norm. We study how well the norm of a typical polynomial on the unit ball of $\\ell_\\infty^n$, namely the cube $[-1,1]^n$, can be recovered from its values at the vertices. For $P\\in\\mathcal{P}_m(\\mathbb{R}^n)$, set \\[ M(P)=\\max_{x\\in[-1,1]^n}|P(x)|, \\qquad V(P)=\\max_{\\varepsilon\\in\\{-1,1\\}^n}|P(\\varepsilon)|. \\] If $P_n$ is chosen according to $γ_{m,n}$, we prove that the relative loss \\[ 1-\\frac{V(P_n)}{M(P_n)} \\] is of order at most $n^{-1/2}$ in probability. Consequently, for every $0<β<1/2$, \\[ γ_{m,n}\\left\\{ P\\in\\mathcal{P}_m(\\mathbb{R}^n): V(P)\\geq (1-n^{-β})M(P) \\right\\} \\longrightarrow1 \\] as $n\\to\\infty$. Thus, with respect to the Bombieri Gaussian measure, the vertices of the cube are asymptotically norming for homogeneous polynomials of fixed degree.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32526",
  "author_total": 2,
  "authors": [
    {
      "name": "Damián Pinasco"
    },
    {
      "name": "Ignacio Zalduendo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8h1wg8c2e7y6h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32526",
  "fulltext_url": null,
  "id": "8h1wg8c2e7y6h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32526",
  "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",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7214785488",
  "pdf_url": "https://arxiv.org/pdf/2609.32526",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8h1wg8c2e7y6h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32526",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32526"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Almost norming vertices for homogeneous polynomials on the cube",
  "title_text": "Almost norming vertices for homogeneous polynomials on the cube",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/8h1wg8c2e7y6h",
  "venue": "arXiv",
  "versions": []
}