Sharp geometric inequalities via optimal mass transportation:
Riemannian versus sub-Riemannian


Motivated by the Pansu conjecture (i.e., the sharp isoperimetric 
inequality in the first Heisenberg group), we present certain sharp
geometric inequalities in the sub-Riemannian setting. By using optimal
mass transportation, we establish Borell-Brascamp-Lieb,
Prékopa-Leindler and Brunn-Minkowski type inequalities on
Heisenberg/Carnot groups. Two approaches are presented: (1)
approximating the sub-Riemannian structures by suitable Riemannian
metrics, cf. Ambrosio-Rigot; (2) intrinsic way (including also the
non-ideal case), cf. Figalli-Rifford. In particular, we shall
emphasize the subtle difference between our results and those by Lott,
Sturm and Villani in the Riemannian setting.