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The sharp quantitative Euclidean concentration inequality. (English) Zbl 1385.39005

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)].

MSC:

39B62 Functional inequalities, including subadditivity, convexity, etc.
49J10 Existence theories for free problems in two or more independent variables

Citations:

Zbl 1187.52009
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