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The convex hull of a uniform sample from the interior of a simple d-polytope

Published online by Cambridge University Press:  14 July 2016

Barthold F. Van Wel
Barthold F. Van Wel*
Affiliation:
University of Amsterdam
*
∗ Postal address: Department of Mathematics, University of Amsterdam, Roetersstraat 15, 1018 WB Amsterdam, The Netherlands.
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Abstract

An asymptotic expression is given for the expected number of vertices of the convex hull of a uniform sample from the interior of a d-dimensional simple polytope. This extends a result derived by Rényi and Sulanke for sample points in the plane.

Keywords

GEOMETRIC PROBABILITYSIMPLE d-POLYTOPESUNIFORM SAMPLE
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 2 , June 1989 , pp. 259 - 273
Copyright
Copyright © Applied Probability Trust 1989 

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References

Brøndsted, A. (1983) An Introduction to Convex Polytopes. Springer-Verlag, New York.Google Scholar
Buchta, C. (1984) Stochastische Approximation konvexer Polygone. Z. Wahrscheinlichkeitsth. 67, 283–304.Google Scholar
Buchta, C. (1985) Zufällige Polyeder-Eine Übersicht. In Zahlentheoretische Analysis, edited by Hlawka, E. Lecture Notes in Mathematics 1114, Springer-Verlag, Berlin, 1–13.Google Scholar
Efron, B. (1965) The convex hull of a random set of points. Biometrika 52, 331–343.Google Scholar
Groeneboom, P. (1987) Limits theorems for convex hulls. Proc. Theory Rel. Fields. Google Scholar
Raynaud, H. (1965) Sur le comportement asymptotique de l'enveloppe convexe d'un nuage de points tirés au hasard dans ℝn . C. R. Acad. Sci. Paris 261, 627–629.Google Scholar
Raynaud, H. (1970) Sur l'enveloppe convexe des nuages de points aléatoires dans ℝn. I. J. Appl. Prob. 7, 35–48.Google Scholar
Renyi, A. and Sulanke, R. (1963), (1964) Über die konvexe Hülle von n zufällig gewählten Punkten, I und IL Z. Wahrscheinlichkeitsth. 2, 75–84; 3, 138–147.Google Scholar
Wieacker, J. A. (1978) Einige probleme der Polyedrischen Approximation. Diplomarbeit, Albert-Ludwigs-Universität, Frieburg im Breisgau.Google Scholar