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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.

References

Daniels, H. E. (1952) The statistical theory of stiff chains. Proc. R. Soc. Edinburgh A63, 290–311.Google Scholar
George, E. I. (1987) Sampling random polygons. J. Appl. Prob., 24, 557–573.CrossRefGoogle Scholar
Kendall, D. G. (1984) Shape manifolds, procrustean metrics, and complex projective spaces. Bull. London Math. Soc. 16, 81–121.Google Scholar
Kingman, J. F. C. (1982) The thrown string. J. R. Statist. Soc. B44, 109–122.Google Scholar
Matheron, G. (1975) Random Sets and Integral Geometry. Wiley, New York.Google Scholar
Miles, R. E. (1961) Random polytopes: the generalisation to n dimensions of the intervals of a Poisson process. Ph.D. thesis, Cambridge University.Google Scholar
Miles, R. E. (1964) Random polygons determined by random lines in a plane. Proc. Nat. Acad. Sci. USA 52, 901–907; 1157-1160.Google Scholar
Miles, R. E. (1969) Poisson flats in Euclidean spaces. Part I: A finite number of random uniform flats. Adv. Appl. Prob. 1, 211–237.Google Scholar
Miles, R. E. (1971) Poisson flats in Euclidean spaces. Part II: Homogeneous Poisson flats and the complementary theorem. Adv. Appl. Prob. 3, 1–43.Google Scholar
Miles, R. E. (1973) The various aggregates of random polygons determined by random lines in a plane. Adv. Math. 10, 256–290.Google Scholar
Miles, R. E. (1986) Random tessellations. Pp. 567–572 in Encyclopedia of Statistical Sciences, Vol. 7. ed. Kotz, S. and Johnson, N. L., Wiley, New York.Google Scholar
Rényi, A. and Sulanke, R. (1968) Züfallige Konvexe Polygone in einem Ringgebiet. Z. Wahrscheinlichkeitsth. 9, 146–157.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1987) Stochastic Geometry and its Applications. Wiley, New York.Google Scholar
Watson, G. N. (1958) A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press.Google Scholar