an:06682219
Zbl 1368.52004
Cuf??, Juli??; Revent??s, Agust??
A lower bound for the isoperimetric deficit
EN
Elem. Math. 71, No. 4, 156-167 (2016).
00361014
2016
j
52A40 51M16 51M25 52A10
plane compact convex sets; isoperimetric inequality; isoperimetric deficit; evolute; pedal curve; Steiner point; Bonnesen-style inequality
Given a plane compact convex set \(K\) of area \(F\) with boundary curve C of length \(L\), the \textit{isoperimetric deficit} is \(\Delta:=L^2-4\pi F\) and the classical \textit{isoperimetric inequality} states \(\Delta \geq 0\), with equality only for discs. As for upper bounds, \textit{A. Hurwitz} [Ann. Sci. ??c. Norm. Sup??r. (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 \textit{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 \textit{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 \textit{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 \textit{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.
Pietro De Poi (Udine)
JFM 33.0599.02