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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Intrinsic volumes of random polytopes with vertices on the boundary of a convex body


Authors: Károly J. Böröczky, Ferenc Fodor and Daniel Hug
Journal: Trans. Amer. Math. Soc. 365 (2013), 785-809
MSC (2010): Primary 52A22; Secondary 60D05, 52A27
Published electronically: June 20, 2012
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Abstract: Let $ K$ be a convex body in $ \mathbb{R}^d$, let $ j\in \{1, \ldots , d-1\}$, and let $ \varrho $ be a positive and continuous probability density function with respect to the $ (d-1)$-dimensional Hausdorff measure on the boundary $ \partial K$ of $ K$. Denote by $ K_n$ the convex hull of $ n$ points chosen randomly and independently from $ \partial K$ according to the probability distribution determined by $ \varrho $. For the case when $ \partial K$ is a $ C^2$ submanifold of $ \mathbb{R}^d$ with everywhere positive Gauss curvature, M. Reitzner proved an asymptotic formula for the expectation of the difference of the $ j$th intrinsic volumes of $ K$ and $ K_n$, as $ n\to \infty $. In this article, we extend this result to the case when the only condition on $ K$ is that a ball rolls freely in $ K$.


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Additional Information

Károly J. Böröczky
Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, 1053 Budapest, Hungary
Email: carlos@renyi.hu

Ferenc Fodor
Affiliation: Department of Geometry, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary – and – Department of Mathematics and Statistics, University of Calgary, 2500 University Dr. N.W., Calgary, Alberta, Canada T2N 1N4
Email: fodorf@math.u-szeged.hu

Daniel Hug
Affiliation: Karlsruhe Institute of Technology, Department of Mathematics, D-76128 Karlsruhe, Germany
Email: daniel.hug@kit.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05648-0
PII: S 0002-9947(2012)05648-0
Received by editor(s): March 23, 2011
Published electronically: June 20, 2012
Additional Notes: The first author was supported by OTKA grant 75016, and by the EU Marie Curie FP7 IEF grant GEOSUMSETS
The second author was supported by Hungarian OTKA grants 68398 and 75016 and by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.
Article copyright: © Copyright 2012 American Mathematical Society