Structure of measures for which Ehrhard symmetrization is perimeter non-increasing
In this paper, we prove that isotropic Gaussian functions are \textit{characterized} by a rearrangement inequality for weighted perimeter in dimensions $n \geq 2$ within the class of non-negative weights in $L^1(\mathbb{R}^n) \cap W^{1,1}_{loc}(\mathbb{R}^n)$. More specifically, we prove that within this class, generalized Ehrhard symmetrization is perimeter non-increasing for all measurable sets in all directions if and only if the distribution function is an isotropic Gaussian. The class of non-negative $L^1(\mathbb{R}^n) \cap W^{1,1}_{loc}(\mathbb{R}^n)$-weights is the broadest class in which this problem can be posed for distributional perimeter. One of the main challenges in this paper is handling these weights without imposing any additional structure. Principally, we establish that generalized Ehrhard symmetrization preserves $μ$-measurability through a novel approximation argument. Additionally, our proof that a rearrangement inequality for weighted perimeter implies that half-spaces are isoperimetric sets is new in the context of generalized Ehrhard symmetrization. Moreover, our version of a variational argument, which had previously appeared in [Rosales, 2014] and [Brock-Chiacchio-Mercaldo, 2008], is carried out under minimal regularity. Finally, we establish some basic but useful results for weighted BV functions with non-negative $L^1(\mathbb{R}^n) \cap W^{1,1}_{loc}(\mathbb{R}^n)$-weights which may be of independent interest.