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# Cambridge Journal of Mathematics

## Volume 6 (2018)

### Number 1

### The sharp quantitative Euclidean concentration inequality

Pages: 59 – 87

DOI: https://dx.doi.org/10.4310/CJM.2018.v6.n1.a3

#### Authors

#### Abstract

The Euclidean concentration inequality states that, among sets with fixed volume, balls have $r$-neighborhoods of minimal volume for every $r \gt 0$. On an arbitrary set, the deviation of this volume growth from that of a ball is shown to control the square of the volume of the symmetric difference between the set and a ball. This estimate is sharp and includes, as a special case, the sharp quantitative isoperimetric inequality proved in “The sharp quantitative isoperimetric inequality” [N. Fusco, F. Maggi, and A. Pratelli, *Ann. Math.,* 168:941–980, 2008].

A. Figalli was supported by NSF Grants DMS-1262411 and DMS-1361122.

F. Maggi was supported by NSF Grants DMS-1265910 and DMS-1361122.

C. Mooney was supported by NSF Grant DMS-1501152.

Received 13 July 2017

Published 27 March 2018