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

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

R. E. Miles
R. E. Miles*
Affiliation:
The Australian National University
*
*Centre for Mathematics and its Applications, Australian National University, Canberra. Postal address: RMB 345, Queanbeyan, NSW 2620, Australia.
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Abstract

In the early 1940s David Kendall conjectured that the shapes of the ‘large' (i.e. large area A) convex polygons determined by a standard Poisson line process in the plane tend to circularity (as A increases). Subject only to one heuristic argument, this conjecture and the corresponding two results with A replaced in turn by number of sides N and perimeter S, are proved. Two further similar limiting distributions are considered and, finally, corresponding limiting non-deterministic shape distributions for the small polygons are determined.

Keywords

RANDOM TESSELLATIONPOISSON LINE PROCESS IN THE PLANELARGE AND SMALL RANDOM POLYGONSASYMPTOTIC SHAPE DISTRIBUTIONSLIMITING CIRCULARITY ANDLINEARITYRANDOM TRIANGULAR SHAPES: DANIELS'STIFF CHAIN THEORY

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 27 , Issue 2 , June 1995 , pp. 397 - 417
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

The original version of this paper was presented at the 18th European Meeting of Statisticians held in Berlin (GDR) on 22–26 August 1988.

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