Starting from a mass transportation proof of the Brunn-Minkowski inequality on convex sets, we improve the inequality showing a sharp estimate about the stability property of optimal sets. This is based on a Poincaré-type trace inequality on convex sets that is also proved in sharp form.

Relative to the Gaussian measure on R, entropy and Fisher information are famously related via Gross’ logarithmic Sobolev inequality (LSI). These same functionals also separately satisfy convolution inequalities, as proved by Stam. We establish a dimension-free inequality that interpolates among these relations. Several interesting corollaries follow: (i) the deficit in the LSI satisfies a conv...

We investigate the validity and stability of various Minkowski-like inequalities for C 1 -perturbations ball. Let K ? R n be a domain (possibly not convex mean-convex) which is -close to prove sharp geometric inequality ( ? ? ? II d H ? ) 2 ? Per , where constant that yields equality when = B (and sum absolute values eigenvalues second fundamental form ). Moreover, any ? > 0 if sufficiently bal...

Abstract Using harmonic mean curvature flow, we establish a sharp Minkowski-type lower bound for total of convex surfaces with given area in Cartan-Hadamard $3$-manifolds. This inequality also improves the known estimates hyperbolic $3$-space. As an application, obtain Bonnesen-style isoperimetric distance function nonpositively curved $3$-spaces, via monotonicity results curvature. connection ...

Abstract. A generalization of Young’s inequality for convolution with sharp constant is conjectured for scenarios where more than two functions are being convolved, and it is proven for certain parameter ranges. The conjecture would provide a unified proof of recent entropy power inequalities of Barron and Madiman, as well as of a (conjectured) generalization of the Brunn-Minkowski inequality. ...