Local minimality of the ball for the Gaussian perimeter (Q1739068)

If $E\subset\mathbb R^n$, then the Gaussian measure is defined as $\gamma(E)=\frac{1}{(2\pi)^{\frac{\pi}{2}}}\int\limits_Ee^{-\frac{|x|^2}2}dx$. Since $\gamma(\mathbb R^n)=1$, the Gauss space $(\mathbb R^n,\gamma)$ is a probability space. For a smooth set $E$, the Gaussian perimeter $P_\gamma(E)$ is defined as $P(E)=\frac{1}{(2\pi)^{\frac{\pi}{2}}}\int\limits_{\partial E}e^{-\frac{|x|^2}2}d \mathcal{H}^{n-1}(x)$. The isoperimetric inequality in Gaussian space states that among all subsets of $R^{n}$ with prescribed Gaussian measure halfspaces have the least Gaussian perimeter, that is, for $s\in\mathbb R$ and the set $H_s=\{x;\ x_n<s\}\subset\mathbb R^n$, the inequality $P_\gamma(E)\ge P_\gamma(H_s)$ holds for all Borel subsets $E\subset\mathbb R^n$ such that $\gamma(E)=\gamma(H_s)$. \par In this paper, the author considers sets $E$ symmetric around the origin, i.e., $E=-E$. He shows that when restricted to this class, balls $B_r$ centered at the origin are local minimizers for the perimeter, at least when $r$ is not too big. The author proves that for $n\ge 2$ if $\sigma\in\left(0,\frac12\right)$, there exist $\delta$ and $\kappa$ such that if $r\in[\sigma,\sqrt{n+1}-\sigma]$, $E$ is a set of locally finite perimeter with $E=-E$, $\gamma(E)=\gamma(B_r)$, and $\gamma(E\Delta B_r)<\delta$, then $P_\gamma(E)-P_\gamma(B_r)\ge\kappa(n,\sigma)\gamma(E\Delta B_r)^2$. In the one-dimensional case, the author shows that $B_r$ is always a local minimizer of the perimeter among symmetric sets with the same Gaussian measure. Moreover, balls are the unique global minimizers for $r>r_0$, where $r_0$ is the unique positive number such that $\frac1{\sqrt{2\pi}}\int\limits_{-r_0}^{r_0}e^{-\frac{t^2}2}dt=\frac12$, while $\mathbb R\backslash B_r$ is the unique global minimizer when $0< r < r_0$.