{
  "abstract": "We study a Poincaré-type inequality for Gaussian measures restricted to origin-symmetric convex sets. The inequality arises from the second variation at $t=0$ of the function $t\\longmapsto γ_Σ\\left(e^{tA}K\\right)$, where $Σ$ and $A$ are symmetric matrices. The relevance of this function comes from Saroglou's reformulation of the log-Brunn--Minkowski conjecture. We show that $Σ$ and $A$ commute if and only if the inequality holds for every box of the form $UD B_\\infty^n$, where $U$ is orthogonal and $D$ is diagonal. In the non-commuting case, this yields counterexamples for orthogonal boxes whose axes can be chosen arbitrarily close to the coordinate axes. We also show that the inequality holds when $K$ is unconditional with respect to an eigenbasis of $Σ$ and $A$ is an arbitrary symmetric matrix.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07248",
  "author_total": 1,
  "authors": [
    {
      "name": "Pavlos Kalantzopoulos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/az1854wptt4at.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07248",
  "fulltext_url": null,
  "id": "az1854wptt4at",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07248",
  "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": "W7220979184",
  "pdf_url": "https://arxiv.org/pdf/2610.07248",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/az1854wptt4at.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07248",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07248"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Commutativity in a Poincaré-Type Inequality",
  "title_text": "Commutativity in a Poincaré-Type Inequality",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/az1854wptt4at",
  "venue": "arXiv",
  "versions": []
}