We will discuss several symmetrization procedures (Steiner, Ehrhard, spherical, and circular symmetrization)
that are known not to increase the perimeter.
First, we will show how it is possible to characterize those sets whose perimeter remains unchanged under symmetrization.
After that, we will also characterize rigidity of equality cases.
By rigidity, we mean the situation when those sets whose perimeter remains unchanged under symmetrization,
are trivially obtained through a rigid motion of the (Steiner, Ehrhard or spherical) symmetral.
We will achieve this through the introduction of a suitable measure-theoretic notion of connectedness,
and through a fine analysis of the barycenter function for a special class of sets.
These results are obtained together with several collaborators
(Maria Colombo, Guido De Philippis, Francesco Maggi, Matteo Perugini, Dominik Stöger).
This is a joint ACA-GAPDE seminar