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The convex hull of a spherically symmetric sample

Published online by Cambridge University Press:  01 July 2016

William F. Eddy  and
James D. Gale
William F. Eddy*
Affiliation:
Carnegie-Mellon University
James D. Gale*
Affiliation:
Carnegie-Mellon University
*
Postal address for both authors: Department of Statistics, Carnegie-Mellon University, Schenley Park, Pittsburgh, PA 15213, U.S.A.
Postal address for both authors: Department of Statistics, Carnegie-Mellon University, Schenley Park, Pittsburgh, PA 15213, U.S.A.
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Abstract

Using the isomorphism between convex subsets of Euclidean space and continuous functions on the unit sphere we describe the probability measure of the convex hull of a random sample. When the sample is spherically symmetric the asymptotic behavior of this measure is determined. There are three distinct limit measures, each corresponding to one of the classical extreme-value distributions. Several properties of each limit are determined.

Keywords

EXTREME-VALUE PROCESSPOISSON RANDOM MEASURERANDOM SETSUPPORT FUNCTIONDIRECTIONWISE MAXIMUM
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 4 , December 1981 , pp. 751 - 763
Copyright
Copyright © Applied Probability Trust 1981 

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Footnotes

∗

Research supported in part by NSF Grants MCS 78-02422 and MCS-80-05115 to Carnegie-Mellon University.

∗∗

Based in part on a portion of this author's Ph.D. thesis submitted to Carnegie-Mellon University.

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