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Structure of measures for which Ehrhard symmetrization is perimeter non-increasing

Sean McCurdy, Kuan-Ting Yeh

TL;DR

The paper characterizes PS measures for generalized Ehrhard symmetrization in $\mathbb{R}^n$ with $n\ge2$: the only finite, nonnegative densities $f$ in $L^1\cap W^{1,1}_{\mathrm{loc}}$ for which $\mathrm{Per}_\mu(S_{\vec{v}}(E))\le \mathrm{Per}_\mu(E)$ for all measurable $E$ and all directions $\vec{v}$ are isotropic Gaussian densities $f(x)=C e^{-c|x-a|^2}$. The authors develop a robust variational framework for weighted BV functions with minimal regularity, extend Ehrhard symmetrization to general measures, prove that half-spaces minimize perimeter, and demonstrate a product structure for the density that, combined with 1D symmetry, yields the Gaussian form after translating to center $a$. They also provide a 1D counterexample showing that the dimension assumption is essential and supply detailed measurability results for the symmetrization map. The work contributes a broad, rigorous characterization of measures supporting perimeter-rearrangement inequalities in Gauss-like settings and strengthens the link between symmetrization, isoperimetry, and measure structure.

Abstract

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.

Structure of measures for which Ehrhard symmetrization is perimeter non-increasing

TL;DR

The paper characterizes PS measures for generalized Ehrhard symmetrization in with : the only finite, nonnegative densities in for which for all measurable and all directions are isotropic Gaussian densities . The authors develop a robust variational framework for weighted BV functions with minimal regularity, extend Ehrhard symmetrization to general measures, prove that half-spaces minimize perimeter, and demonstrate a product structure for the density that, combined with 1D symmetry, yields the Gaussian form after translating to center . They also provide a 1D counterexample showing that the dimension assumption is essential and supply detailed measurability results for the symmetrization map. The work contributes a broad, rigorous characterization of measures supporting perimeter-rearrangement inequalities in Gauss-like settings and strengthens the link between symmetrization, isoperimetry, and measure structure.

Abstract

In this paper, we prove that isotropic Gaussian functions are \textit{characterized} by a rearrangement inequality for weighted perimeter in dimensions within the class of non-negative weights in . 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 -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 -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 -weights which may be of independent interest.
Paper Structure (13 sections, 23 theorems, 214 equations, 1 figure)

This paper contains 13 sections, 23 theorems, 214 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and preliminaries
  3. Sets of finite $\mu$-perimeter and weighted BV functions
  4. Generalized Ehrhard symmetrization
  5. Half-planes are isoperimetric sets
  6. Proof of Lemma \ref{['l:H are minimal']}
  7. The density f enjoys a product structure
  8. Proof of Theorem \ref{['t:main theorem 2']}
  9. Appendix A: An example in one dimension
  10. Appendix B: Measurability
  11. Proof of Theorem \ref{['l:measurability']} (assuming Lemma \ref{['l:meas implies meas arbitrary II']})
  12. Proof of Lemma \ref{['l:meas implies meas arbitrary I']}
  13. Proof of Lemma \ref{['l:meas implies meas arbitrary II']}

Key Result

Theorem 1.3

Let $n \ge 2$ and $\mu \in \mathscr{W}(\mathbb{R}^n)$, as above. Then, for all measurable sets $E \subset \mathbb{R}^n$ and all $\vec{v}\in \mathbb{S}^{n-1}$ if and only if $\mu$ has a distribution function $f(x) = Ce^{-c|x-a|^2}$ for some $0<c<\infty$, $0 \le C <\infty$, and $a \in \mathbb{R}^n$, where $S_{\vec{v}}$ is the generalized $\vec{v}$-Ehrhard symmetrization

Figures (1)

  • Figure 1: Moving mass from $E \setminus H_{\mu}(E, \vec{v})$ to "fill in" $H_{\mu}(E, \vec{v}) \setminus E$.

Theorems & Definitions (53)

  • Definition 1.2: The class of measures
  • Theorem 1.3: Main Theorem
  • Theorem 1.4: Isotropic Uniqueness
  • Remark 1.5
  • Definition 2.1: Weighted $BV$ functions
  • Lemma 2.2: Riesz Representation Theorem
  • Lemma 2.3: Lower Semicontinuity
  • proof
  • Lemma 2.4: Weak approximation by smooth functions has bounded weighted variation
  • proof
  • ...and 43 more