A simplified proof of a conjecture of D. G. Kendall concerning shapes of random polygons. (1999) - Kovalenko, Igor N.
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 → ∞ .