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A lower bound for the isoperimetric deficit. (English) Zbl 1368.52004
Given a plane compact convex set \(K\) of area \(F\) with boundary curve C of length \(L\), the isoperimetric deficit is \(\Delta:=L^2-4\pi F\) and the classical isoperimetric inequality states \(\Delta \geq 0\), with equality only for discs. As for upper bounds, A. Hurwitz [Ann. Sci. Éc. Norm. Supér. (3) 19, 357–408 (1902; JFM 33.0599.02)] proved that \(0\leq \Delta \leq \pi |F_e|\), where \(F_e\) is the algebraic area encloded by the evolute (i.e. the locus of the centres of curvature) of \(C\).
For lower bounds, during the 1920’s, T. Bonnesen proved a series of inequalities of the form \(\Delta\geq B\), where \(B\) has the following three basic properties: it is non-negative; it can vanish only when \(C\) is a circle; \(B\) has geometric significance. A Bonnesen-style inequality is an inequality as above which satisfies the three basic properties.
In the paper under review, the authors prove a Bonnesen-style inequality; to state this, we need some definitions: the pedal curve of \(C\) with respect to a fixed point \(O\) is the locus of points \(X\) so that the line \(\overline{OX}\) is perpendicular to the tangent to \(C\) passing through \(X\); the Steiner point of \(K\) is the centre of mass of \(C\) with respect to the density function that assigns to each point of \(C\) its curvature. Let \(A\) be the area enclosed by the pedal curve with respect to the Steiner point of \(K\); then, in Theorem 3.1 the authors prove that \(\Delta\geq 3\pi (A-F)\). Moreover, the authors improve the above inequality in special cases, and consider also when the equality holds.

MSC:
52A40 Inequalities and extremum problems involving convexity in convex geometry
51M16 Inequalities and extremum problems in real or complex geometry
51M25 Length, area and volume in real or complex geometry
52A10 Convex sets in \(2\) dimensions (including convex curves)
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References:
[1] C.A. Escudero and A. Revent’os. An interesting property of the evolute. Amer. Math. Monthly, 114(7):623– 628, 2007. · Zbl 1144.53007
[2] H. Groemer. Geometric applications of Fourier series and spherical harmonics, volume 61 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1996. · Zbl 0877.52002
[3] A. Hurwitz. Sur quelques applications g’eom’etriques des s’eries de Fourier. Annales scientifiques de l’ ’E.N.S., 19(3e s’erie):357–408, 1902.
[4] Robert Osserman. Bonnesen-style isoperimetric inequalities. Amer. Math. Monthly, 86(1):1–29, 1979. · Zbl 0404.52012
[5] L.A. Santal’o. Integral Geometry and Geometric Probability. Cambridge University Press, 2004. Second edition.
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