Robust dimension free isoperimetry in Gaussian space

TL;DR

This paper proves a dimension-free stability result for the Gaussian isoperimetric inequality, showing that sets nearly achieving the boundary measure bound are close to half-spaces, with explicit bounds independent of dimension.

Contribution

It provides the first dimension-free stability estimate for Gaussian isoperimetry with explicit bounds, improving upon previous results that depended on the dimension.

Findings

Establishes a logarithmic inverse relation between boundary measure excess and set proximity to a half-space.

Achieves dimension-free bounds, unlike prior results with dimension-dependent constants.

Demonstrates that near-optimal boundary measure implies closeness to a half-space with explicit quantitative bounds.

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$.