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

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.

11 pages, 2 figures. With respect to the previous versions we changed the title and we explicitly computed some constants

References in corpus (3)

• A two-sided estimate for the Gaussian noise stability deficit
• Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality
• Near Equality in the Brunn-Minkowski Inequality

Cited by in corpus (2)

• Quantitative isoperimetry à la Levy-Gromov
• Quantitative lower bounds to the Euclidean and the Gaussian Cheeger constants