6533b7cffe1ef96bd1258671
RESEARCH PRODUCT
Robustness of the Gaussian concentration inequality and the Brunn–Minkowski inequality
Vesa JulinMarco Barchiesisubject
Pure mathematicsGaussianConvex setkvantitatiivinen tutkimus01 natural sciencesMeasure (mathematics)Square (algebra)010104 statistics & probabilitysymbols.namesakeMathematics - Analysis of PDEsQuantitative Isoperimetric InequalitiesFOS: MathematicsMathematics::Metric Geometry0101 mathematicsConcentration inequalitySymmetric differenceMathematicsmatematiikkaApplied MathematicsProbability (math.PR)010102 general mathematicsMinkowski inequalityMinkowski additionBrunn–Minkowski inequalityGaussian concentration inequalitysymbols49Q20 52A40 60E15Mathematics - ProbabilityAnalysisAnalysis of PDEs (math.AP)description
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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2016-08-29 | Calculus of Variations and Partial Differential Equations |