Dimensional variance inequalities of Brascamp-Lieb type and a local approach to dimensional Prékopaʼs theorem

We give a new approach, inspired by Hörmanderʼs L2-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 Hölder inequalities for convex functions, weighted Brascamp-Lieb inequalities and sharp weighted Poincaré inequalities for generalized Cauchy measures.