The Gaussian Conjugate Rogers-Shephard Inequality
Emanuel Milman, Shohei Nakamura, Hiroshi Tsuji
TL;DR
This work proves a sharp Gaussian conjugate Rogers–Shephard inequality by merging the Rogers–Shephard–Spingarn framework with Royen’s Gaussian correlation inequality through a novel Gaussian Forward-Reverse Brascamp–Lieb (FRBL) inequality for centered log-concave functions, accommodating degenerate covariances. The main result, GCRSI, states $\gamma(K) \gamma(L) \leq \gamma(K \cap L) \gamma(K + L)$ for convex sets with Gaussian barycenters at the origin, with equality cases fully characterized and several conjugate or functional variants developed. A key technical engine is the FRBL inequality, which enables Gaussian saturation arguments and yields both Gaussian and Lebesgue-form consequences, including CRSSI and GCMPI-type results. The paper also provides a functional formulation and a unified viewpoint that connects geometric inequalities to functional ones via max_0 and square-convolution, offering a broad methodological framework and several open directions for extending saturation techniques. Overall, the results extend classical Rogers–Shephard–Milman-type inequalities into the Gaussian setting with sharp constants and equality structure, and they propose a robust, functional approach to geometric-analytic inequalities with potential for wide applicability.
Abstract
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.