Robust dimension free isoperimetry in Gaussian space

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