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Identities linking volumes of convex hulls

Part of:Geometric probability and stochastic geometryCombinatorial probability

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 

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