{
  "abstract": "Let $\\boldsymbol{X} = (X_1,\\ldots,X_n)$ be a centered Gaussian vector, not necessarily nondegenerate. It is proved that, for every $α_1,\\ldots,α_n > 0$, \\[ \\mathsf{E}\\left[\\prod_{i=1}^n |X_i|^{α_i}\\right] \\geq \\prod_{i=1}^n \\mathsf{E}\\left[|X_i|^{α_i}\\right]. \\] When all marginal variances are positive, equality holds if and only if the coordinates are independent. This settles Frenkel's 18-year-old Gaussian product inequality (GPI) conjecture and, in fact, its later strengthening to arbitrary positive exponents. Through Frenkel's hafnian formulation, this result also provides a short proof of the 28-year-old real linear polarization constant conjecture, which was settled this year as a consequence of the strong polarization inequality. The main result leads to the exact real linear polarization constant, a sharp weighted product inequality for real linear functionals, spherical moment bounds, hafnian inequalities for positive-semidefinite matrices together with complete characterizations of their equality cases, logarithmic variance and covariance inequalities, a Rényi total-correlation certificate, and unconditional versions of mixed-sign Gaussian moment bounds previously conditional on the positive-exponent GPI.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20234",
  "author_total": 2,
  "authors": [
    {
      "name": "Frédéric Ouimet"
    },
    {
      "name": "Dylan Greaves"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6edzfy4pgvg2p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20234",
  "fulltext_url": null,
  "id": "6edzfy4pgvg2p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20234",
  "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"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7213590968",
  "pdf_url": "https://arxiv.org/pdf/2609.20234",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6edzfy4pgvg2p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20234",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20234"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "A proof of the strong Gaussian product inequality conjecture",
  "title_text": "A proof of the strong Gaussian product inequality conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:29Z",
  "url": "https://research.draconitic.com/paper/6edzfy4pgvg2p",
  "venue": "arXiv",
  "versions": []
}