Robust dimension free isoperimetry in Gaussian space

Abstract

We prove the first robust dimension free isoperimetric result for the standard Gaussian measure $\gamma_{n}$ and the corresponding boundary measure $\gamma_{n}^{+}$ in $\mathbb{R} ^{n}$. The main result in the theory of Gaussian isoperimetry (proven in the 1970s by Sudakov and Tsirelson, and independently by Borell) states that if $\gamma_{n}(A)=1/2$ then the surface area of $A$ is bounded by the surface area of a half-space with the same measure, $\gamma_{n}^{+}(A)\leq(2\pi)^{-1/2}$. Our results imply in particular that if $A\subset\mathbb{R} ^{n}$ satisfies $\gamma_{n}(A)=1/2$ and $\gamma_{n}^{+}(A)\leq(2\pi)^{-1/2}+\delta$ then there exists a half-space $B\subset\mathbb{R} ^{n}$ such that $\gamma_{n}(A\Delta B)\leq C\smash{\log^{-1/2}}(1/\delta)$ for an absolute constant $C$. Since the Gaussian isoperimetric result was established, only recently a robust version of the Gaussian isoperimetric result was obtained by Cianchi et al., who showed that $\gamma_{n}(A\Delta B)\le C(n)\sqrt{\delta}$ for some function $C(n)$ with no effective bounds. Compared to the results of Cianchi et al., our results have optimal (i.e., no) dependence on the dimension, but worse dependence on $\delta$.