BZ Seminar in Analysis: Francesco Maggi (University of Texas, Austin)

Rigidity of equality cases in symmetrization inequalities

In a seminal paper by Chleb“k, Cianchi and Fusco (Ann. Math. 2005), a sufficient condition for rigidity of equality cases in Steiner inequality was presented. (By rigidity of equality cases we mean the situation when all sets realizing equality in the Steiner's inequality defined by a given symmetric set are necessarily symmetric.) Their condition is however far from being necessary, and the problem of characterizing rigidity of equality cases was left open even when the given symmetric set is a polyhedron.
In this talk we first introduce a measure-theoretic notion of connectedness, inspired by Federer's notion of indecomposable current, that is then exploited to prove several characterization results of rigidity of equality cases, both in the case of the classical Steiner's inequality, as well as in the case of Ehrhard's inequality for Gaussian perimeter. This is a joint work with Filippo Cagnetti (U. Sussex), Maria Colombo (SNS Pisa), and Guido De Philippis (U. Bonn).

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