{
  "abstract": "We study the maximizers of the generalized volume product \\[ γ_σ^n(A)\\,γ_σ^n(A^\\circ) \\] among all measurable subsets $A\\subset\\mathbb{R}^n$, where $A^\\circ$ denotes the polar set of $A$, and where $γ_σ^n$ denotes the centered Gaussian probability measure on $\\mathbb{R}^n$ with covariance $σ^2 I_n$, $σ>0$. It turns out that the maximizers depend on $σ$. We prove that they exist and are convex bodies. In dimension $n=1$, we find the exact form of the maximizers. In dimension $n\\ge 2$, we show that they are smooth bodies of revolution whose support function satisfies a certain differential equation. Moreover, for $σ^2 \\le \\frac{1}{n}$ we show that the Euclidean unit ball is the unique maximizer, while this is no longer the case for $σ^2\\ge {\\frac{2}{n+1}}$. In dimension $n=2$, we close the gap by showing that the Euclidean unit ball is the unique maximizer for $σ^2 \\le \\frac{2}{3}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18472",
  "author_total": 3,
  "authors": [
    {
      "name": "S. Artstein-Avidan"
    },
    {
      "name": "M. Fradelizi"
    },
    {
      "name": "K. Wyczesany"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9d6997cckcjnk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18472",
  "fulltext_url": null,
  "id": "9d6997cckcjnk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18472",
  "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.MG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.18472",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9d6997cckcjnk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18472",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18472"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Uncentered Blaschke-Santaló inequalities for the Gaussian measure",
  "title_text": "Uncentered Blaschke-Santaló inequalities for the Gaussian measure",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/9d6997cckcjnk",
  "venue": "arXiv",
  "versions": []
}