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Robust dimension free isoperimetry in Gaussian space

We prove the first robust dimension free isoperimetric result for the standard Gaussian measure $γ_n$ and the corresponding boundary measure $γ_n^+$ 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 $γ_n(A)=1/2$ then the surface area of $A$ is bounded by the surface area of a half-space with the same measure, $γ_n^+(A)\leq(2π)^{-1/2}$. Our results imply in particular that if $A\subset \mathbb {R}^n$ satisfies $γ_n(A)=1/2$ and $γ_n^+(A)\leq(2π)^{-1/2}+δ$ then there exists a half-space $B\subset \mathbb {R}^n$ such that $γ_n(AΔB)\leq C\smash{\log^{-1/2}}(1/δ)$ 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 $γ_n(AΔB)\le C(n)\sqrtδ$ 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 $ δ$.