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Sharp quantitative isoperimetric inequalities in the \(L^1\) Minkowski plane. (English) Zbl 1207.52007

The isoperimetric inequality in the Euclidean \(n\)-space \(\mathbb{R}^n\) states that if \(E\) is any Borel set, then \(|\delta (E)| \geq n \omega_n^{1/n} |E|^{(n-1)/n}\), with equality if and only if \(E\) is a ball. Here and throughout, \(\delta (E)\) is the boundary of \(E\), \(| \dots |\) is the (Lebesgue) measure of the appropriate dimension, and \(\omega_n\) is the measure of the unit ball. If one calls the quantity \(\left(|\delta (E)| / \left[n \omega_n^{1/n} |E|^{(n-1)/n} \right] \right) -1\) the isoperimetric deficit of \(E\) and denotes it by \(D(E)\), then one expects that if \(D(E)\) is small, then \(E\) is close to being a ball. An interesting quantitative version of this last statement, using the Fraenkel asymmetry \(\lambda (E)\) of \(E\) as a measure of how far from being a ball \(E\) is, is established by N. Fusco, F. Maggi, and A. Pratelli in [Ann. Math. (2), 168, No. 3, 941–980 (2008; Zbl 1187.52009)], where it is proved that \(\lambda (E) \leq \gamma(n) (D(E))^{1/2}\), where \(\gamma(n)\) is a constant that depends on \(n\) only, and where the exponent 1/2 cannot be improved upon. This statement was proved for the exponent 1/4 and was conjectured for 1/2 by R. R. Hall in [J. Reine Angew. Math. 428, 161–176 (1992; Zbl 0746.52012)]. Here, the Fraenkel asymmetry \(\lambda (E)\) of \(E\) is defined by \(\lambda (E) = \inf \{ |E~ \Delta ~B| / |E| : B \text{~is a ball with~} |B|=|E|\}\), where \(\Delta\) denotes the symmetric difference.
The paper under review establishes quantitative isoperimetric inequalities in the Minkowski \(L_1\)-plane \(\mathbb{M}\), i.e., the plane \(\mathbb{R}_2\) with the \(L_1\)-norm. In \(\mathbb{M}\), the isoperimetric inequality states that if \(E\) is a domain and if \(\|\delta (E)\|\) denotes the \(L_1\)-length of \(\delta (E)\), then \(\|\delta (E)\|^2 \geq 16 |E|\), with equality if and only if \(E\) is a square, where a square is to stand for one with sides parallel to the coordinate axes. For a measure of the isoperimetric deficit of \(E\), the author seems to consider the quantity \(D(E) = \left(\|\delta (E)\|^2 / 16 |E|\right) - 1\) (which is essentially the same as \(\left(\|\delta (E)\| / 4 \sqrt{|E|}\right) - 1\)). For a measure of the asymmetry of \(E\) (with respect to the square \(S\)), he (implicitly) considers two different measures \(\lambda_H\) and \(\lambda_F\). The first is defined by \(\lambda_H(E) = \inf \{d_{\infty} (E,S)^2/|E| : S \text{~is a square}\},\) where \(d_{\infty}\) is the \(L^{\infty}\) Hausdorff metric defined on the set of compact sets by \(d_{\infty} (E,F) = \inf \{ t \geq 0 : E \subseteq F + B_{\infty} (t) \text{~and~} E \subseteq F + B_{\infty} (t)\},\) and where \(B_{\infty} (t)\) is the ball of radius \(t\) in the \(L^{\infty}\) norm. The second is defined, as \(\lambda\) was defined earlier, by \(\lambda_F (E) = \inf \{ |E~ \Delta ~ S| / |E| : S \text{~is a square with~} |S| = |E|\}\). He then proves the following quantitative isoperimetric inequalities: 6,5mm
(i)
if \(D(E) \leq \varepsilon\), then \(\lambda_H (E) \leq \varepsilon / 64\),
(ii)
if \(D(E) \leq \varepsilon\), then \(\lambda_F (E) \leq \sqrt{\varepsilon} / 2 + O(\varepsilon)\) (for \(\varepsilon\) near 0).

MSC:

52A40 Inequalities and extremum problems involving convexity in convex geometry
49Q20 Variational problems in a geometric measure-theoretic setting
51M25 Length, area and volume in real or complex geometry
51M16 Inequalities and extremum problems in real or complex geometry
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References:

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