  Characterization results for equality cases and for rigidity of equality
cases in Steiner's perimeter inequality are presented. (By rigidity, we mean
the situation when all equality cases are vertical translations of the
Steiner's symmetral under consideration.) We achieve this through the
introduction of a suitable measure-theoretic notion of connectedness and a fine
analysis of barycenter functions for sets of finite perimeter having segments
as orthogonal sections with respect to an hyperplane.
