Intrinsic volumes of random polytopes with vertices on the boundary of a convex body
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- by Károly J. Böröczky, Ferenc Fodor and Daniel Hug PDF
- Trans. Amer. Math. Soc. 365 (2013), 785-809 Request permission
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 $K$.References
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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
- MR Author ID: 619845
- Email: fodorf@math.u-szeged.hu
- Daniel Hug
- Affiliation: Karlsruhe Institute of Technology, Department of Mathematics, D-76128 Karlsruhe, Germany
- MR Author ID: 363423
- Email: daniel.hug@kit.edu
- 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. - © Copyright 2012 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 365 (2013), 785-809
- MSC (2010): Primary 52A22; Secondary 60D05, 52A27
- DOI: https://doi.org/10.1090/S0002-9947-2012-05648-0
- MathSciNet review: 2995373