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The sharp quantitative isoperimetric inequality. (English) Zbl 1187.52009
The isoperimetric inequality states that for any Borel set \(E\subset{\mathbb R}^n\), \(n\geq 2\), with finite Lebesgue measure \(|E|\) it holds \(P(E)\geq n\omega_n^{1/n}|E|^{(n-1)/n}\), with equality if and only if \(E\) is a ball. Here \(P\) denotes the (distributional) perimeter and \(\omega_n\) is the measure of the unit ball \(B\subset{\mathbb R}^n\).
In this remarkable paper the authors prove a quantitative sharp form of the classical isoperimetric inequality. More precisely, let \(D(E)\) denote the isoperimetric deficit of \(E\), i.e., \[ D(E)=\frac{P(E)}{n\omega_n^{1/n}|E|^{(n-1)/n}}-1, \] and let \(\lambda(E)\) be the so-called Fraenkel asymmetry of \(E\), which is defined by \[ \lambda(E)=\min\left\{\frac{d(E,x+rB)}{r^n}:x\in{\mathbb R}^n\right\}, \] where \(r>0\) is such that \(|rB|=|E|\) and \(d(E,F)=|E\Delta F|\) denotes the measure of the symmetric difference between the Borel sets \(E\) and \(F\). In [J. Reine Angew. Math. 428, 161–176 (1992; Zbl 0746.52012)], R. Hall proved that \(\lambda(E)\leq C(n)D(E)^{1/4}\), where \(C(n)\) is a constant depending only on \(n\). Moreover, he conjectured that the term \(D(E)^{1/4}\) was not optimal and it should be replaced by the smaller one \(D(E)^{1/2}\).
In the paper under review the authors settle Hall’s conjecture, by showing that for any Borel set \(E\subset{\mathbb R}^n\), \(n\geq 2\), with \(0<|E|<\infty\), there exists a constant \(C(n)\) such that \[ \lambda(E)\leq C(n)D(E)^{1/2}. \] The power \(1/2\) is optimal in any dimension.

52A40 Inequalities and extremum problems involving convexity in convex geometry
28A75 Length, area, volume, other geometric measure theory
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