The Anisotropic Gaussian Isoperimetric Inequality and Ehrhard Symmetrization

The symmetrization method is a captivating tool for handling isoperimetric-type inequalities, with numerous applications across various fields. In this talk, we provide an overview of topics pertaining to isoperimetric-type inequalities and symmetrizations, focusing on the perspective of geometric measure theory. We demonstrate inequalities associated with symmetrizations and highlight that within a broad class of measures, the isotropic Gaussian measure is the sole measure that satisfies the perimeter-decreasing property.