Bonnesen-style inequalities and pseudo-perimeters for polygons
Let \(C\) be a simple closed curve in the Euclidean plane \(E\) with perimeter \(L\). Let \(A\) denote the area of the bounded component of \(E \setminus C\). The classical isoperimetric inequality states that \(L^2 - 4\pi A \geq 0\) holds with equality if and only if \(C\) is a circle. This inequality can be generalized to the so-called Bonnesen inequality \(L^2 - 4\pi A \geq \pi^2(R-r)^2\), where \(R\) is the infimum of all radii of circles that contain \(C\) and \(r\) is the supremum of radii of circles that are contained in \(C\) [see \textit{R. Osserman}, Am. Math. Mon. 86, 1-29 (1979; Zbl 0404.52012), Complex Variables 9, 241-249 (1987; Zbl 0602.52006), and Bull. Am. Math. Soc. 84, 1182-1238 (1978; Zbl 0411.52006)]. In the paper under review the author considers Bonnesen-style isoperimetric inequalities of the form \(L^2 - 4\pi A \geq B\), where \(B \geq 0\), \(B=0\) if and only if \(C\) is a circle, and \(B\) has some geometric interpretation. Here, \(C\) is a cyclic \(n\)-gon rather than a smooth simple curve. An \(n\)-gon is called cyclic if all of its vertices lie on some fixed circle. The author proves five Bonnesen-style inequalities for cyclic \(n\)-gons \(C\) such as Theorem A. Let \(C\) be a cyclic \(n\)-gon inscribed in a circle \(K\) and let \(l\) denote the perimeter of the regular \(n\)-gon inscribed in \(K\). Then \(L^2 - n\tan(\pi/n)A \geq (l-L)^2\) with equality if and only if \(C\) is a regular \(n\)-gon. Theorem D. Let \(C\) be a cyclic \(n\)-gon inscribed in a circle \(K\). Let \(H\) be the total altitude of \(C\), i.e. \(H\) is the sum of all distances from the centroid of \(P\) to each of its sides. Let \(h\) denote the total altitude of the regular \(n\)-gon inscribed in \(K\). Then \(H^2 - n\cot(\pi/n)A \geq (h - H)^2\) with equality if and only if \(C\) is a regular \(n\)-gon.
scientific article; zbMATH DE number 1106611
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