Figalli, Alessio; Maggi, Francesco; Mooney, Connor The sharp quantitative Euclidean concentration inequality. (English) Zbl 1385.39005 Camb. J. Math. 6, No. 1, 59-87 (2018). Summary: 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 “The sharp quantitative isoperimetric inequality” [N. Fusco et al., Ann. Math. (2) 168, No. 3, 941–980 (2008; Zbl 1187.52009)]. Cited in 10 Documents MSC: 39B62 Functional inequalities, including subadditivity, convexity, etc. 49J10 Existence theories for free problems in two or more independent variables Keywords:Euclidean concentration inequality; minimal volume; isoperimetric inequality Citations:Zbl 1187.52009 PDFBibTeX XMLCite \textit{A. Figalli} et al., Camb. J. Math. 6, No. 1, 59--87 (2018; Zbl 1385.39005) Full Text: DOI arXiv