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Limit theorems for random polytopes with vertices on convex surfaces

Part of: Limit theorems Geometric probability and stochastic geometry General convexity

Published online by Cambridge University Press:  29 November 2018

N. Turchi  and
F. Wespi
N. Turchi*
Affiliation:
Ruhr University Bochum
F. Wespi*
Affiliation:
University of Bern
*
* Postal address: Faculty of Mathematics, Ruhr University Bochum, Universitätsstrasse 150, 44780 Bochum, Germany. Email address: nicola.turchi@ruhr-uni-bochum.de
** Postal address: Department of Mathematics and Statistics, University of Bern, Sidlerstrasse 150, 3012 Bernv, Switzerland.
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Abstract

We consider the random polytope Kn, defined as the convex hull of n points chosen independently and uniformly at random on the boundary of a smooth convex body in ℝd. We present both lower and upper variance bounds, a strong law of large numbers, and a central limit theorem for the intrinsic volumes of Kn. A normal approximation bound from Stein's method and estimates for surface bodies are among the tools involved.

Keywords

Central limit theoremintrinsic volumerandom polytopestochastic geometrysurface bodyvariance

MSC classification

Primary: 52A22: Random convex sets and integral geometry
Secondary: 60D05: Geometric probability and stochastic geometry 60F05: Central limit and other weak theorems
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 4 , December 2018 , pp. 1227 - 1245
Copyright
Copyright © Applied Probability Trust 2018 

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