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The probability that two samples in the plane will have disjoint convex hulls

Published online by Cambridge University Press:  14 July 2016

L. C. G. Rogers
L. C. G. Rogers*
Affiliation:
University of Cambridge
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Abstract

Suppose given an absolutely continuous distribution on the plane, and points Pi, · · ·, Pj, Πt, · · ·, Πk chosen independently according to the given distribution. Denoting by Gj the convex hull of {Pi, · · ·, Pj}, and by Γk the convex hull of (Πt, · · ·, Πk}, and writing pjk for the probability that Gj and Γk are disjoint, certain properties of the array {pjk; j, k = 1,2, · · ·} are established, including a recurrence generating the array in terms of {p1n; n = 1,2, · · ·}, and asymptotic results for {pnn; n = 1,2, · · ·}. Some examples are considered.

Keywords

BIVARIATE NORMAL DISTRIBUTIONCLOCKWISE CRITICAL LINECONVEX HULL
Type
Research Papers
Information
Journal of Applied Probability , Volume 15 , Issue 4 , December 1978 , pp. 790 - 802
Copyright
Copyright © Applied Probability Trust 1978 

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References

Carnal, H. (1970) Die konvexe Hülle von n rotationsymmetrisch verteilten Punkten. Z. Wahrscheinlichkeitsth. 15, 168–179.Google Scholar
Efron, B. (1965) The convex hull of a random set of points. Biometrika 52, 331–343.Google Scholar
Kendall, D. G. (1977) Computer techniques and the archival map-reconstruction of Mycenaean Messenia. In Mycenaean Geography 1976 , ed. Bintliff, J. University Library Press, Cambridge.Google Scholar
Rényi, A. and Sulanke, R. (1963) Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitsth. 2, 75–84.Google Scholar
Ruben, H. (1954) On the moments of order statistics in samples from normal populations. Biometrika 41, 200–227.Google Scholar