A simplified proof of a conjecture of D. G. Kendall concerning shapes of random polygons
type: Item
instance of: scholarly article
title: A simplified proof of a conjecture of D. G. Kendall concerning shapes of random polygons
zbMATH Open document ID: 0959.60007
DOI: 10.1155/S1048953399000283
author: I. M. Kovalenko
published in: Journal of Applied Mathematics and Stochastic Analysis
publication date: 2001-04-26T00:00:00Z
full work available at URL: https://eudml.org/doc/48702
review text: 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 \(\lambda\to\infty\).
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 1452748
zbMATH Keywords: stochastic geometry
zbMATH Keywords: random sets
zbMATH Keywords: random tessellation
Wikidata QID: Q21147003
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1452748