Gaussian inequalities with different covariance matrices - the Heterovariant Gaussian Brunn-Minkowski inequality

We establish the following novel heterovariant Gaussian Brunn-Minkowski inequality (involving general covariance matrices ${\mathrm{\Sigma }}_1,{\mathrm{\Sigma }}_2,\mathrm{\Gamma }$) for origin-symmetric convex sets $K,L\subset {\mathbb{R}}^n$ and $p,\lambda \in (0,1)$: $${\gamma }_{{\mathrm{\Sigma }}_1}(K{)}^{1-p}{\gamma }_{{\mathrm{\Sigma }}_2}(L{)}^p\le {\left({\left(\frac{1-p}{1-\lambda }\right)}^{1-p}{\left(\frac{p}{\lambda }\right)}^p\right)}^n{\gamma }_{\mathrm{\Gamma }}((1-\lambda )K+\lambda L),$$ whenever $\mathrm{\Gamma }\le {\mathrm{\Sigma }}_1{\mathrm{♯}}_p{\mathrm{\Sigma }}_2$, the $p$-geometric mean of ${\mathrm{\Sigma }}_1,{\mathrm{\Sigma }}_2$. This immediately implies both the dimensional Gaussian Brunn-Minkowski inequality of Eskenazis-Moschidis (by using ${\mathrm{\Sigma }}_1={\mathrm{\Sigma }}_2=\mathrm{\Gamma }$ and optimizing over $p$) and the strong Gaussian (B)-theorem of Cordero-Erausquin-Fradelizi-Maurey (by using $K=L$, $p=\lambda$ 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 $\lambda \in [0,1]$: $$\gamma (aK{)}^{1-\lambda }\gamma (bL{)}^{\lambda }\le \gamma \left(a^{1-\lambda }{b}^{\lambda }((1-\lambda )K+\lambda L)\right)\mathrm{.}$$ 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.