Cambridge Journal of Mathematics

Volume 6 (2018)

Number 1

The sharp quantitative Euclidean concentration inequality

Pages: 59 – 87



Alessio Figalli (Department of Mathematics, ETH Zürich, Switzerland)

Francesco Maggi (Department of Mathematics, University of Texas, Austin, Tx., U.S.A.)

Connor Mooney (Department of Mathematics, ETH Zürich, Switzerland)


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