Journal of Applied Mathematics and Stochastic Analysis
Volume 12 (1999), Issue 4, Pages 301-310
doi:10.1155/S1048953399000283
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→∞.