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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.

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