The Brunn-Minkowski inequality and the Heisenberg group

The core of this thesis is the Brunn-Minkowski inequality. We start proving the general version of the Brunn-Minkowski inequality in the Euclidean space, then we investigate the validity of a "geodesic" version of the Brunn-Minkowski inequality in the Heisenberg group endowed with its sub-Riemannian distance. We provide geometric evidence of the exponents involved in the generalized Brunn-Minkowski inequality, by discussing the so called Measure Contraction Property in the Heisenberg group.