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 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 a hyperplane. 1. Introduction 1535 2. Notions from geometric measure theory 1549 3. Characterization of equality cases and barycenter functions 1555 4. Rigidity in Steiner's inequality 1579 Appendix A. Equality cases in the localized Steiner inequality 1589 Appendix B. A perimeter formula for vertically convex sets 1591 Acknowledgements 1592 References
