Geometric inequalities on Heisenberg groups

We establish geometric inequalities in the sub-Riemannian setting of the Heisenberg group $\mathbb H^n$. Our results include a natural sub-Riemannian version of the celebrated curvature-dimension condition of Lott-Villani and Sturm and also a geodesic version of the Borell-Brascamp-Lieb inequality akin to the one obtained by Cordero-Erausquin, McCann and Schmuckenschl\"ager. The latter statement implies sub-Riemannian versions of the geodesic Pr\'ekopa-Leindler and Brunn-Minkowski inequalities. The proofs are based on optimal mass transportation and Riemannian approximation of $\mathbb H^n$ developed by Ambrosio and Rigot. These results refute a general point of view, according to which no geometric inequalities can be derived by optimal mass transportation on singular spaces.