Robustness of the Gaussian concentration inequality and the Brunn-Minkowski inequality

TL;DR

This paper establishes sharp quantitative stability results for the Gaussian concentration inequality and the Brunn-Minkowski inequality, linking set enlargements to geometric proximity to half-spaces or convex sets.

Contribution

It provides the first sharp stability estimates for these fundamental inequalities, extending to general convex sets in Euclidean space.

Findings

Quantitative stability of Gaussian concentration inequality.

Stability estimate for Euclidean Brunn-Minkowski inequality.

Control of symmetric difference by set enlargements.

Abstract

We provide a sharp quantitative version of the Gaussian concentration inequality: for every $r>0$, the difference between the measure of the $r$-enlargement of a given set and the $r$-enlargement of a half-space controls the square of the measure of the symmetric difference between the set and a suitable half-space. We also prove a similar estimate in the Euclidean setting for the enlargement with a general convex set. This is equivalent to the stability of the Brunn-Minkowski inequality for the Minkowski sum between a convex set and a generic one.