The sharp quantitative Euclidean concentration inequality

2016/01/16 by Figalli, Alessio, Maggi, Francesco, Mooney, Connor · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1601.04100

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 \citefuscomaggipratelli.

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• Sharp gradient stability for a class of Hardy-Sobolev-Maz'ya inequalities
• Sharp quantitative integral inequalities for harmonic extensions

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