The sharp quantitative Euclidean concentration inequality

TL;DR

This paper establishes a sharp quantitative version of the Euclidean concentration inequality, linking set deviations to volume growth and connecting to isoperimetric and Riesz rearrangement inequalities.

Contribution

It provides the first sharp quantitative form of the Euclidean concentration inequality, relating set deviations to volume growth and applications to isoperimetric and Riesz inequalities.

Findings

Proves a sharp quantitative bound relating set deviation to volume growth.

Shows the inequality's implications for the Euclidean isoperimetric inequality.

Connects the concentration inequality to Riesz rearrangement inequalities.

Abstract

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 from \cite{fuscomaggipratelli}.