By systematically dualizing a generalized version of the Reilly formula (an integrated form of Bochner’s formula in the presence of boundary), we obtain new Poincar ́e-type inequalities on a Riemannian manifold equipped with a density and on its boundary under a Curvature-Dimension condition, for various combinations of boundary conditions of the domain (convex, mean-convex) and the function (Neumann, Dirichlet). In particular, we extend and refine the Poincar’e inequalities of Lichnerowicz, Brascamp–Lieb, Bobkov–Ledoux, Nguyen and Colesanti to the weighted Rieman- nian setting, in a single unified framework. We then propose a new geometric evolution equation, which extends to the Riemannian setting the Minkowski addition operation of convex domains, a notion previously confined to the linear setting, and for which a novel Brunn–Minkowski inequality in the weighted-Riemannian setting is obtained. Our framework allows to encompass the entire class of Borell’s convex measures, including heavy-tailed measures, and extends the latter class to weighted-manifolds having negative “dimension”. Based on joint work with Alexander Kolesnikov.http://cvgmt.sns.it/seminar/413/
