Journal of Applied Mathematics and Stochastic Analysis
Volume 12 (1999), Issue 4, Pages 301-310
doi:10.1155/S1048953399000283
A simplified proof of a conjecture of D. G. Kendall concerning shapes of random polygons
National Academy of Sciences of Ukraine, V.M. Glushkov Institute of Cybernetics, Kyiv 252027, Ukraine
Received 1 January 1998; Revised 1 January 1999
Copyright © 1999 Igor N. Kovalenko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Following investigations by Miles, the author has given a few proofs of a
conjecture of D.G. Kendall concerning random polygons determined by the
tessellation of a Euclidean plane by an homogeneous Poisson line process.
This proof seems to be rather elementary. Consider a Poisson line process
of intensity λ on the plane ℛ2 determining the tessellation of the plane
into convex random polygons. Denote by Kω a random polygon containing the origin (so-called Crofton cell). If the area of Kω is known to equal
1, then the probability of the event {the contour of Kω lies between two
concentric circles with the ratio 1+ϵ of their ratio} tends to 1 as λ→∞.