On the Gaussian Projection Inequalities

Abstract We provide an overview of projection bodies in Gaussian probability space for sets of finite Gaussian perimeter and their corresponding applications in functions of Bounded variation space. On the one hand, we study the properties of Gaussian projection bodies for sets of finite Gaussian perimeter under Ehrhard symmetrization and establish a Gaussian projection-type inequality. This inequality concludes that Ehrhard symmetrization contracts the Minkowski symmetrization of the Gaussian projection body for set of finite Gaussian perimeter $E$. On the other hand, we investigate the functional “lifting” of Ehrhard symmetrization and establish the affine Gaussian Pólya–Szegö-type inequalities in terms of the functional Ehrhard symmetrization.