{
  "abstract": "We establish the following novel heterovariant Gaussian Brunn-Minkowski inequality (involving general covariance matrices $Σ_1,Σ_2,Γ$) for origin-symmetric convex sets $K,L \\subset \\mathbb{R}^n$ and $p,λ\\in (0,1)$: \\[ γ_{Σ_1}(K)^{1-p} γ_{Σ_2}(L)^p \\le \\left( \\left( \\frac{1-p}{1-λ} \\right)^{1-p} \\left( \\frac{p}λ \\right)^p \\right)^{n} γ_Γ((1-λ) K + λL) , \\] whenever $Γ\\leq Σ_1 \\sharp_p Σ_2$, the $p$-geometric mean of $Σ_1,Σ_2$. This immediately implies both the dimensional Gaussian Brunn-Minkowski inequality of Eskenazis-Moschidis (by using $Σ_1 = Σ_2 = Γ$ and optimizing over $p$) and the strong Gaussian (B)-theorem of Cordero-Erausquin-Fradelizi-Maurey (by using $K=L$, $p=λ$ and diagonal covariances), fusing them into a single unified statement. In fact, the inequality applies to convex sets whose appropriate Gaussian barycenter is at the origin. Furthermore, it reveals a new *bilinear* version of the strong Gaussian (B)-theorem (involving *two* convex sets); in particular, for all $a,b > 0$ and $λ\\in [0,1]$: \\[ γ(a K)^{1-λ} γ(b L)^λ \\leq γ\\left ( a^{1-λ} b^λ\\left ((1-λ) K + λL \\right) \\right ) . \\] A more general heterovariant version of the (bilinear) (B)-theorem replaces dilation by general linear transformations. A different version yields a heterovariant form of the recently established Gaussian conjugate Rogers-Shephard inequality. By averaging these Gaussian inequalities with respect to the covariance matrices, we obtain dimensional Brunn-Minkowski and conjugate Rogers-Shephard inequalities for Gaussian mixtures. The proofs are based on the Gaussian Forward-Reverse Brascamp-Lieb inequality established in our previous work.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13596",
  "author_total": 3,
  "authors": [
    {
      "name": "Emanuel Milman"
    },
    {
      "name": "Shohei Nakamura"
    },
    {
      "name": "Hiroshi Tsuji"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1rw18e1pd0zja.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13596",
  "fulltext_url": null,
  "id": "1rw18e1pd0zja",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13596",
  "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"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7221576330",
  "pdf_url": "https://arxiv.org/pdf/2610.13596",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1rw18e1pd0zja.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.13596",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13596"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Gaussian inequalities with different covariance matrices - the Heterovariant Gaussian Brunn-Minkowski inequality",
  "title_text": "Gaussian inequalities with different covariance matrices - the Heterovariant Gaussian Brunn-Minkowski inequality",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/1rw18e1pd0zja",
  "venue": "arXiv",
  "versions": []
}