The sharp quantitative Euclidean concentration inequality

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 estimate is sharp and includes, as a special case, the sharp quantitative isoperimetric inequality proved in [FMP08].