Cambridge Journal of Mathematics
Volume 6 (2018)
The sharp quantitative Euclidean concentration inequality
Pages: 59 – 87
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