  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.
