{
  "abstract": "Talagrand asked for a construction of a large convex subset of a bounded Minkowski sum of a large Gaussian set. We first prove a stronger nonsymmetric existential statement: if $A\\subset\\mathbb{R}^n$ is measurable and $γ_n(A)>2/3$, then $A+A+A$ contains a compact convex set of Gaussian measure at least $1/2$. Let now $A$ be closed with $γ_n(A)\\ge7/8$, let $Φ$ be the standard Gaussian distribution function, and put $a_0=Φ^{-1}(3/4)$. For every $Λ>1$ and every $0<p<2Φ(Λa_0)-1$ we construct a centrally symmetric finite polytope $C\\subsetΛ(A+A+A)$ with $γ_n(C)\\ge p$. Thus measure $3/4$ is obtained for every $Λ>1.705510\\ldots$. We give matching upper and lower bounds for the optimal high-measure dilation and show that the dilation profile used by the construction is sharp among all symmetric half-measure cores. Finally, if $γ_n(A)\\ge5/6+η$, we construct, for every $0<\\varepsilon\\le1/2$, a centered ellipsoid $E_\\varepsilon$ with $γ_n(E_\\varepsilon)\\ge1-\\varepsilon$ and \\begin{equation} \\frac{c}{Φ^{-1}(1-\\varepsilon/2)}\\sqrt{\\frac{\\log n}{n}}\\,E_\\varepsilon\\subset A+A+A. \\end{equation} Both the dimension dependence and the dependence on $\\varepsilon$ are optimal up to constants. The construction also gives a six-summand theorem for balanced sets and a second finite construction based on subgaussian tests.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14832",
  "author_total": 1,
  "authors": [
    {
      "name": "Othmane Mazhar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5c714yezfy3z9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14832",
  "fulltext_url": null,
  "id": "5c714yezfy3z9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14832",
  "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.MG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7213301149",
  "pdf_url": "https://arxiv.org/pdf/2609.14832",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5c714yezfy3z9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.14832",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14832"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A constructive solution to Talagrand's Gaussian convexification problem",
  "title_text": "A constructive solution to Talagrand's Gaussian convexification problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/5c714yezfy3z9",
  "venue": "arXiv",
  "versions": []
}