{
  "abstract": "Let $u$ be the torsion function for the Ornstein-Uhlenbeck operator on a bounded domain $Ω\\subset \\mathbb{R}^n$, i.e., the solution of $Δu - x \\cdot \\nabla u = -1$ in $Ω$ with $u = 0$ on $\\partialΩ$. Let $T_γ(Ω) = \\int_Ωu \\, dγ$ be the Gaussian torsional rigidity. We prove that the Brunn-Minkowski-type inequality $T_γ((1-t)Ω_0 + tΩ_1)^α \\le (1-t) T_γ(Ω_0)^α + t T_γ(Ω_1)^α$ fails for every exponent $α> 0$ and in every dimension $n \\ge 2$, for a pair of convex bodies centrally symmetric with respect to the origin, which may be taken smooth with positive curvature. This answers Conjecture 1.4 for any $n \\geq 2$, and Question~(Q), of Marín Sola and Salerno in the negative. The mechanism is a first-order lower bound for $T_γ$ at a ball $Ω_0$ under Minkowski perturbations. When the perturbing body $Ω_1$ has small torsion and large mean width, $T_γ((1-t)Ω_0 + tΩ_1)$ increases to first order. Since the Minkowski combination has larger torsion than both endpoints, no exponent can repair the inequality. For $n=1$, convexity with the optimal exponent $1/3$ holds on symmetric intervals by results of the same authors, \\cite{MSS26}. We prove that the logarithm of the torsion is neither convex nor concave along Minkowski combinations of symmetric intervals, that no non-zero exponent yields concavity, and that convexity fails for every positive exponent when one set is a union of two intervals or when the sets are reflected off-center intervals.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29941",
  "author_total": 2,
  "authors": [
    {
      "name": "Xuan Hien Nguyen"
    },
    {
      "name": "Alina Stancu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ah3062eyz30p2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29941",
  "fulltext_url": null,
  "id": "ah3062eyz30p2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29941",
  "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.AP"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7204957236",
  "pdf_url": "https://arxiv.org/pdf/2608.29941",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ah3062eyz30p2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2608.29941",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29941"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Failure of a Brunn-Minkowski-type inequality for the Gaussian torsional rigidity",
  "title_text": "Failure of a Brunn-Minkowski-type inequality for the Gaussian torsional rigidity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/ah3062eyz30p2",
  "venue": "arXiv",
  "versions": []
}