Gaussian inequalities with different covariance matrices - the Heterovariant Gaussian Brunn-Minkowski inequality
Abstract
We establish the following novel heterovariant Gaussian Brunn-Minkowski inequality (involving general covariance matrices Σ1,Σ2,Γ) for origin-symmetric convex sets K,L ⊂ ℝn and p,λ∈ (0,1): γΣ1(K)1−p γΣ2(L)p ≤ ( ( (1−p)/(1−λ) )1−p ( p/λ )p )n γΓ((1−λ) K + λL) , whenever Γ≤ Σ1 ♯p Σ2, the p-geometric mean of Σ1,Σ2. This immediately implies both the dimensional Gaussian Brunn-Minkowski inequality of Eskenazis-Moschidis (by using Σ1 = Σ2 = 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∈ [0,1]: γ(a K)1−λ γ(b L)λ ≤ γ ( a1−λ bλ ((1−λ) K + λL ) ) . 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.
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BibTeX
@misc{milman2026gaussian,
title = {{Gaussian inequalities with different covariance matrices - the Heterovariant Gaussian Brunn-Minkowski inequality}},
author = {Emanuel Milman and Shohei Nakamura and Hiroshi Tsuji},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13596},
eprint = {2610.13596},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13596},
}
RIS
TY - UNPB
TI - Gaussian inequalities with different covariance matrices - the Heterovariant Gaussian Brunn-Minkowski inequality
AU - Emanuel Milman
AU - Shohei Nakamura
AU - Hiroshi Tsuji
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.13596
UR - https://arxiv.org/abs/2610.13596
AB - 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 ⊂ ℝ^n and p,λ∈ (0,1): γ_Σ_1(K)^1−p γ_Σ_2(L)^p ≤ ( ( (1−p)/(1−λ) )^1−p ( p/λ )^p )^n γ_Γ((1−λ) K + λL) , whenever Γ≤ Σ_1 ♯_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∈ [0,1]: γ(a K)^1−λ γ(b L)^λ ≤ γ ( a^1−λ b^λ ((1−λ) K + λL ) ) . 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.
ER -