We fuse between the Rogers-Shephard inequality for the Lebesgue measure and Royen's Gaussian Correlation Inequality, simultaneously extending both into a single sharp inequality for the Gaussian measure $γ$ on $\mathbb{R}^n$, stating that \[ γ(K) γ(L) \leq γ(K\cap L) γ(K+L) \] whenever $K$ and $L$ are origin-symmetric convex sets in $\mathbb{R}^n$. This confirms a conjecture of M. Tehranchi [https://doi.org/10.1214/17-ECP89]. In fact, we show that the inequality remains valid whenever the Gaussian barycenters of $K$ and $L$ are at the origin, and characterize the equality cases. After rescaling, this also yields the following new inequality for convex sets with (Lebesgue) barycenters at the origin: \[ |K| |L| \leq |K \cap L| |K + L | ; \] this can be seen as a conjugate counterpart to Spingarn's extension of the Rogers-Shephard inequality (where $K+L$ is replaced by $K-L$ above). We also derive an additional conjugate version of a Gaussian inequality due to V. Milman and Pajor, as well as several extensions. Our main tool is a new Gaussian Forward-Reverse Brascamp-Lieb inequality for centered log-concave functions, of independent interest, which is crucially applicable to degenerate Gaussian covariances.

arXiv

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Other Link： https://arxiv.org/pdf/2602.07981v1

Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

Abstract

Motivated by the barycenter problem in optimal transportation theory, Kolesnikov–Werner recently extended the concept of the Legendre duality relation from two functions to multiple functions. We further generalize this duality relation and then establish the centered Gaussian saturation property for a Blaschke–Santaló-type inequality associated with it. Our approach to understanding this generalized Legendre duality relation is based on the observation that directly links Legendre duality with the inverse Brascamp–Lieb inequality. More precisely, for a large family of degenerate Brascamp–Lieb data, we prove that the centered Gaussian saturation property for the inverse Brascamp–Lieb inequality holds when inputs are restricted to even and log-concave functions. As an application to convex geometry, we establish a significant case of a conjecture by Kolesnikov–Werner concerning the Blaschke–Santaló inequality for multiple even functions and multiple symmetric convex bodies. Furthermore, in the context of information theory and optimal transportation theory, this provides an affirmative answer to another conjecture by Kolesnikov–Werner concerning a Talagrand-type inequality for multiple even probability measures involving the Wasserstein barycenter.

DOI： 10.1007/s00222-025-01382-5

arXiv

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Other Link： https://link.springer.com/article/10.1007/s00222-025-01382-5/fulltext.html

Publishing type：Research paper (scientific journal)   Publisher：European Mathematical Society - EMS - Publishing House GmbH  

We prove that the functional volume product for even functions is increasing along the Fokker–Planck heat flow. This in particular yields a new proof of the functional Blaschke–Santaló inequality by K. Ball and Artstein-Avidan–Klartag–Milman in the even case. This result is a consequence of a new understanding of the regularizing property of the Ornstein–Uhlenbeck semigroup. That is, we establish an improvement of Borell’s reverse hypercontractivity inequality for even functions and identify the sharp range of the admissible exponents. As another consequence of successfully identifying the sharp range for the inequality, we derive a sharp L^{p} - L^{q} inequality for the Laplace transform for even functions. The best constant of the inequality is attained by centered Gaussians, which provides an analogous result to Beckner’s sharp Hausdorff–Young inequality. Our technical novelty in the proof is the use of the Brascamp–Lieb inequality for log-concave measures and Cramér–Rao’s inequality in this context.

DOI： 10.4171/jems/1720

arXiv

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Completely positive operators are fundamental objects in quantum information
theory and the capacity of such operators plays a pivotal role in the operator
scaling algorithm. Using this algorithm, Garg--Gurvits--Oliveira--Wigderson
recently established a certain quantitative continuity result for the capacity
map at rational points. We show by different means that operator capacity
possesses significantly greater regularity. Our argument gives local H\"{o}lder
continuity at all points and rests on a result of
Bennett--Bez--Buschenhenke--Cowling--Flock on weighted sums of exponential
functions.

arXiv

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Other Link： http://arxiv.org/pdf/2508.02118v1

Publishing type：Research paper (scientific journal)   Publisher：Elsevier BV  

DOI： 10.1016/j.aim.2025.110161

arXiv

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