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A heuristic proof of a long-standing conjecture of D. G. Kendall concerning the shapes of certain large random polygons. (English) Zbl 0829.60008

Let \({\mathcal L}\) denote the standard Poisson line process of intensity \(\tau\) in the plane which (in the obvious way) determines a random (convex polygonal) tessellation \({\mathcal P}\) of \(R^2\). Almost surely, the origin of \(R^2\) lies in a well-defined polygon \(^0P \in {\mathcal P}\). Let \(N,S,A,D\), and \(W\), respectively, denote the number of vertices (= number of sites), the perimeter, the area, the in-diameter, and the width (in a given direction) of \(^0P\). Finally let \(\sigma\) denote the shape of \(^0P\) in the sense of D. G. Kendall [Bull. Lond. Math. Soc. 16, 81-121 (1984; Zbl 0579.62100)]. A long-standing conjecture of D. G. Kendall says that the conditional distribution of \(\sigma\) – given \(A\) – converges weakly to the degenerate law concentrated on a circle as \(A \to \infty\). This is written suggestively as (i) \(\sigma |A \approx \text{disc}\). Similar conjectures are (ii) \(\sigma |N \approx \text{disc}\); (iii) \(\sigma |S \approx \text{disc}\); (iv) \(\sigma |D \approx \text{disc}\), and (v) \(\sigma |W \approx\) line segment (orthogonal to the projection direction). The author proves (iv) and shows that (ii)\(\Rightarrow\)(iii)\(\Rightarrow\)(i). Furthermore he gives (only!) heuristic proofs of (ii) and (v).
Reviewer: K.Schürger (Bonn)

MSC:

60D05 Geometric probability and stochastic geometry

Citations:

Zbl 0579.62100
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