Rigidity for perimeter inequalities under symmetrisation

My PhD thesis contains a couple of results I obtained under the supervision of my advisor Filippo Cagnetti, during the past three years of my studies. In particular, I present two results about rigidity of perimeter inequality under symmetrization techniques. The first result, presented in Chapter 3, provides the characterization of rigidity for equality cases for the perimeter inequality under spherical symmetrization; whereas in Chapter 4 I will study the rigidity of equality cases for the Steiner's inequality for the anisotropic perimeter.