Dimensional variance inequalities of Brascamp-Lieb type and a local approach to dimensional Pr\'ekopa's theorem

TL;DR

This paper introduces a novel local approach inspired by H"ormander's method to establish weighted variance inequalities, leading to new proofs and applications in convex geometry and measure inequalities.

Contribution

It provides a new local proof technique for dimensional Brunn-Minkowski inequalities and extends variance inequalities with multiple applications.

Findings

New local proof of dimensional Brunn-Minkowski inequalities

Extension of variance inequalities to various measures

Derivation of sharp weighted Poincaré inequalities

Abstract

We give a new approach, inspired by H\"ormander's $L^2$-method, to weighted variance inequalities which extend results obtained by Bobkov and Ledoux. It provides in particular a local proof of the dimensional functional forms of the Brunn-Minkowski inequalities. We also present several applications of these variance inequalities, including reverse H\"older inequalities for convex functions, weighted Brascamp-Lieb inequalities and sharp weighted Poincar\'{e} inequalities for generalized Cauchy measures.