Andrea Cianchi, Università di Firenze
The perimeter inequality for Steiner symmetrization: cases of equality

Abstract: A very classical result, going back to Steiner, asserts that the perimeter of any set in the Euclidean space may not increase under symmetrization about any hyperplane. We characterize the sets whose perimeter is unchanged after such a symmetrization. In particular, we find minimal conditions ensuring that the relevant sets are symmetric about the hyperplane of symmetrization.