Inserted: 16 jan 2016
Last Updated: 10 aug 2016
The Euclidean concentration inequality states that, among sets with fixed volume, balls have r-neighborhoods of minimal volume for every r > 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 sharp result is strictly related to the physically significant problem of understanding near maximizers in the Riesz rearrangement inequality with a strictly decreasing radially decreasing kernel. Moreover, it implies as a particular case the sharp quantitative Euclidean isoperimetric inequality proved by Fusco, Maggi and Pratelli.