A simplified proof of a conjecture of D. G. Kendall concerning shapes of random polygons
Summary: Following investigations by \textit{R. E. Miles} [Adv. Appl. Probab. 27, No. 2, 397-417 (1995; Zbl 0829.60008)], the author gives a few proofs of a conjecture of D. G. Kendall concerning random polygons determined by the tessellation of a Euclidean plane by a homogeneous Poisson line process. This proof seems to be rather elementary. Consider a Poisson line process of intensity \(\lambda\) on the plane \({\mathcal R}^2\) determining the tessellation of the plane into convex random polygons. Denote by \(K_\omega\) a random polygon containing the origin (so-alled Crofton cell). If the area of \(K_\omega\) is known to equal 1, then the probability of the event \{the contour of \(K_\omega\) lies between two concentric circles with the ratio \(1+\varepsilon\) of their ratio\} tends to 1 as \(\lambda\to\infty\).
scientific article; zbMATH DE number 1452748