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

Rigidity of equality cases in Steiner's perimeter inequality