Anisotropic Gaussian Measures and Ehrhard Symmetrization

The Gaussian isoperimetric problem holds significant relevance across various fields, with Ehrhard symmetrization serving as a crucial tool in its examination. One natural question to ask is if we can extend those concepts to anisotropic Gaussian measures. In this talk, we uncover an example highlighting a scenario where Ehrhard symmetrization fails to decrease the anisotropic Gaussian perimeter, thereby introducing a novel inequality that incorporates an error term. Finally, we present that within specific classes, only the isotropic Gaussian measure satisfies the perimeter-decreasing property under symmetrization.