Robustness of the Gaussian concentration inequality and the Brunn–Minkowski inequality
Barchiesi, M., & Julin, V. (2017). Robustness of the Gaussian concentration inequality and the Brunn–Minkowski inequality. Calculus of Variations and Partial Differential Equations, 56 (3), 80. doi:10.1007/s00526-017-1169-x
© Springer-Verlag Berlin Heidelberg 2017. This is a final draft version of an article whose final and definitive form has been published by Springer. Published in this repository with the kind permission of the publisher.
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.