The Anisotropic Gaussian Isoperimetric Inequality and Ehrhard Symmetrization

Abstract

In this thesis, we establish the isoperimetric inequality for the anisotropic Gaussian measure and characterize the cases of equality. Additionally, we present an example demonstrating that Ehrhard symmetrization fails to decrease for the anisotropic Gaussian perimeter and introduce a new inequality that includes an error term. This new inequality, in particular, provides a clue to a uniqueness result for the Ehrhard measure within the class of anisotropic Gaussian measures. Our final result, a collaboration with Sean McCurdy, expands the class of measures to which the previous uniqueness result applies.