16 jun 2014 -- 15:05 [open in google calendar]

**Abstract.**

Isoperimetric problems are of great interest both in the Euclidean and in the Gaussian setting. A fundamental tool to understand them is the idea of symmetrizations (in the Euclidean setting, the idea is due to Steiner), as it is known that the perimeter behaves monotonically under symmetrization. Natural questions which arise in this context are then the analysis of equality cases and of rigidity for the symmetrization inequalities, where by rigidity we mean the situation when the equality cases are symmetric sets by themselves. We study a geometric characterization of rigidity. The condition is formulated in terms of a new measure-theoretic notion of connectedness for Borel sets, inspired by Federer?s definition of indecomposable current. (joint work with Filippo Cagnetti, Guido De Philippis, and Francesco Maggi)