A simplified proof of a conjecture of D. G. Kendall concerning shapes of random polygons. (1999)

• Record Type: Journal Article
• Authors: Kovalenko, Igor N.
• Is Part Of: Journal of applied mathematics and stochastic analysis. Volume 12:Number 4(1999)
• Page Start: 301
• Page End: 310
• Publication Date: 1999
• Subjects: stochastic geometry -- random sets -- random tessellation; Mathematical models -- Periodicals; Computer simulation -- Periodicals; Computer science -- Mathematics -- Periodicals; Computer science -- Mathematics; Computer simulation; Mathematical models; Applied Mathematics; Periodicals; Electronic journals; 519.22
• DOI: 10.1155/S1048953399000283
• Languages: English
• ISSNs: 1048-9533

Abstract

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 ℛ 2 determining the tessellation of the plane into convex random polygons. Denote by K ω a random polygon containing the origin (so-called Crofton cell ). If the area of K ω is known to equal 1, then the probability of the event {the contour of K ω lies between two concentric circles with the ratio 1 + ϵ of their ratio} tends to 1 as λ → ∞ .