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

[摘要] 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 intensityλon the planeℛ2determining the tessellation of the plane into convex random polygons. Denote byKωa random polygon containing the origin (so-calledCrofton cell). If the area ofKωis known to equal1, then the probability of the event {the contour ofKωlies between two concentric circles with the ratio1+ϵof their ratio} tends to1asλ→∞.

[学科分类] 应用数学