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.
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Communicated by L.Ambrosio.
This work has been supported by the FiDiPro project “Quantitative Isoperimetric Inequalities” and the Academy of Finland grant 268393.
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Barchiesi, M., Julin, V. Robustness of the Gaussian concentration inequality and the Brunn–Minkowski inequality. Calc. Var. 56, 80 (2017). https://doi.org/10.1007/s00526-017-1169-x
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DOI: https://doi.org/10.1007/s00526-017-1169-x