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Limit theorems for random polytopes with vertices on convex surfaces

Part of: Geometric probability and stochastic geometry Limit theorems 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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References

[1]Bárány, I. and Larman, D. G. (1988).Convex bodies, economic cap coverings, random polytopes.Mathematika 35,274–291.Google Scholar
[2]Bárány, I.,Fodor, F. and Vígh, V. (2010).Intrinsic volumes of inscribed random polytopes in smooth convex bodies.Adv. Appl. Prob. 42,605–619.Google Scholar
[3]Böröczky, K. J.,Fodor, F. and Hug, D. (2013).Intrinsic volumes of random polytopes with vertices on the boundary of a convex body.Trans. Amer. Math. Soc. 365,785–809.Google Scholar
[4]Böröczky, K. J.,Fodor, F.,Reitzner, M. and Vígh, V. (2009).Mean width of random polytopes in a reasonably smooth convex body.J. Multivariate Anal. 100,2287–2295.Google Scholar
[5]Buchta, C.,Müller, J. and Tichy, R. F. (1985).Stochastical approximation of convex bodies.Math. Ann. 271,225–235.Google Scholar
[6]Calka, P. and Yukich, J. E. (2014).Variance asymptotics for random polytopes in smooth convex bodies.Prob. Theory Relat. Fields 158,435–463.Google Scholar
[7]Calka, P. and Yukich, J. E. (2015).Variance asymptotics and scaling limits for {G}aussian polytopes.Prob. Theory Relat. Fields 163,259–301.Google Scholar
[8]Calka, P. and Yukich, J. E. (2017).Variance asymptotics and scaling limits for random polytopes.Adv. Math. 304,1–55.Google Scholar
[9]Calka, P.,Schreiber, T. and Yukich, J. E. (2013).Brownian limits, local limits and variance asymptotics for convex hulls in the ball.Ann. Prob. 41,50–108.Google Scholar
[10]Lachièze-Rey, R. and Peccati, G. (2017).New Berry-Esseen bounds for functionals of binomial point processes.Ann. Appl. Prob. 27,1992–2031.Google Scholar
[11]Lachièze-Rey, R.,Schulte, M. and Yukich, J. (2017). Normal approximation for stabilizing functionals. Preprint. Available at https://arxiv.org/abs/1702.00726v1.Google Scholar
[12]Reitzner, M. (2002).Random points on the boundary of smooth convex bodies.Trans. Amer. Math. Soc. 354,2243–2278.Google Scholar
[13]Reitzner, M. (2003).Random polytopes and the Efron-Stein jackknife inequality.Ann. Prob. 31,2136–2166.Google Scholar
[14]Reitzner, M. (2004).Stochastic approximation of smooth convex bodies.Mathematika 51,11–29.Google Scholar
[15]Reitzner, M. (2005).Central limit theorems for random polytopes.Prob. Theory Relat. Fields 133,483–507.Google Scholar
[16]Richardson, R. M.,Vu, V. H. and Wu, L. (2008).An inscribing model for random polytopes.Discrete Comput. Geom. 39,469–499.Google Scholar
[17]Schneider, R. (2014).Convex Bodies: The Brunn-Minkowski Theory (Encyclopedia Math. Appl. 151).Cambridge University Press.Google Scholar
[18]Schneider, R. and Weil, W. (2008).Stochastic and Integral Geometry.Springer,Berlin.Google Scholar
[19]Schütt, C. and Werner, E. (2003).Polytopes with vertices chosen randomly from the boundary of a convex body. In Geometric Aspects of Functional Analysis (Lecture Notes Math. 1807),Springer,Berlin, pp. 241–422.Google Scholar
[20]Stemeseder, J. (2014).Random polytopes with vertices on the sphere. Doctoral Thesis, University of Salzburg.Google Scholar
[21]Thäle, C. (2018).Central limit theorem for the volume of random polytopes with vertices on the boundary.Discrete Comput. Geom. 59,990–1000.Google Scholar
[22]Thäle, C.,Turchi, N. and Wespi, F. (2018).Random polytopes: central limit theorems for intrinsic volumes.Proc. Amer. Math. Soc. 146,3063–3071.Google Scholar
[23]Vu, V. H. (2005).Sharp concentration of random polytopes.Geom. Funct. Anal. 15,1284–1318.Google Scholar
[24]Vu, V. (2006).Central limit theorems for random polytopes in a smooth convex set.Adv. Math. 207,221–243.Google Scholar