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

created by barchiesi on 29 Aug 2016
modified on 31 Aug 2017

[BibTeX]

Published Paper

Inserted: 29 aug 2016
Last Updated: 31 aug 2017

Journal: Calc. Var. Partial Differential Equations
Volume: 56
Number: 3
Pages: 1-12
Year: 2017
Doi: 10.1007/s00526-017-1169-x
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 provide a similar estimate in the Euclidean case 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.