Search

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. [ABSTRACT FROM AUTHOR]

For sets of finite perimeter, k -codimensional projection bodies are introduced. The corresponding k -codimensional projection inequalities are investigated and their equality cases are characterized. [ABSTRACT FROM AUTHOR]

The affine Orlicz Pólya–Szegö principles with respect to Steiner symmetrization and Schwarz symmetrization proved in Lin and Xing (International mathematics research notices, 2019) are extended to B V (Ω) , the space of functions of bounded variation defined in an open set Ω ⊂ R n , and their equality cases are characterized. [ABSTRACT FROM AUTHOR]