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

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  • Igor N. Kovalenko

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 → ∞ .

Suggested Citation

  • Igor N. Kovalenko, 1999. "A simplified proof of a conjecture of D. G. Kendall concerning shapes of random polygons," International Journal of Stochastic Analysis, Hindawi, vol. 12, pages 1-10, January.
    • Handle: RePEc:hin:jnijsa:195318
    DOI: 10.1155/S1048953399000283
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