Robust Dimension Free Isoperimetry in Gaussian Space

We prove the first robust dimension free isoperimetric result for the standard Gaussian measure γn and the corresponding boundary measure γ+n in Rn. The main result in the theory of Gaussian isoperimetry (proven in the 1970s by Sudakov and Tsirelson, and independently by Borell) states that if γn(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) ≤ (2π)−1/2. Our results imply in particular that if A ⊂ Rn satisfies γn(A) = 1/2 and γ+n(A) ≤ (2π)−1/2+δ then there exists a half-space B ⊂ Rn such that γn(AΔB) ≤ Clog−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 γn(AΔB) ≤ C(n)√δ 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.