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Identities linking volumes of convex hulls

Part of: Combinatorial probability Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Richard Cowan
Richard Cowan*
Affiliation:
University of Sydney
*
∗ Postal address: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia. Email address: rcowan@mail.usyd.edu.au
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Abstract

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Let n points be randomly and independently placed in R d according to a common probability law. It is known that the expected volume for the convex hull of these points, in the cases where n - d ≥ 2 and even, is related linearly to expected volumes of the convex hulls for j points, j < n. We show that similar identities for these volumes hold almost surely - and in contexts where independence and communality of law do not apply. New geometric and topological identities developed here provide a foundation for this result.

Keywords

Convex hullSylvester's problemrandom geometry

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60C05: Combinatorial probability

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 39 , Issue 3 , September 2007 , pp. 630 - 644
Copyright
Copyright © Applied Probability Trust 2007 

References

Affentranger, F. (1988). Generalization of a formula of C. Buchta about the convex hull of random points. Elem. Math. 43, 39–45, 151–152.Google Scholar
Baddeley, A. J. and Møller, J. (1989). Nearest-neighbour Markov point processes and random sets. Internat. Statist. Rev. 57, 89–121.Google Scholar
Baddeley, A. J. and van Lieshout, M. N. M. (1995). Area-interaction point processes. Ann. Inst. Statist. Math. 47, 601–619.Google Scholar
Badertscher, E. (1989). An explicit formula about the convex hull of random points. Elem. Math. 44, 104–106.Google Scholar
Buchta, C. (1986). On a conjecture of R. E. Miles about the convex hull of random points. Monatsh. Math. 102, 91–102.Google Scholar
Buchta, C. (1990). Distribution-independent properties of the convex hull of random points. J. Theoret. Prob. 3, 387–393.Google Scholar
Grünbaum, B. (1967). Convex Polytopes. John Wiley, London.Google Scholar
Miles, R. E. (1971). Isotropic random simplices. Adv. Appl. Prob. 3, 353–382.Google Scholar
Strauss, D. J. (1975). A model for clustering. Biometrika 63, 467–475.Google Scholar