Expectation of intrinsic volumes of random polytopes

Let K be a convex body in ℝ d , let j ∈ {1, …, d −1}, and let K ( n ) be the convex hull of n points chosen randomly, independently and uniformly from K . If ∂ K is C + 2 , then an asymptotic formula is known due to M. Reitzner (and due to I. Bárány if ∂ K is C + 3 ) for the difference of the j th i...

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Bibliographic Details
Published in:Periodica mathematica Hungarica 2008-12, Vol.57 (2), p.143-164
Main Authors: Böröczky, Károly J., Hoffmann, Lars Michael, Hug, Daniel
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
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Summary:Let K be a convex body in ℝ d , let j ∈ {1, …, d −1}, and let K ( n ) be the convex hull of n points chosen randomly, independently and uniformly from K . If ∂ K is C + 2 , then an asymptotic formula is known due to M. Reitzner (and due to I. Bárány if ∂ K is C + 3 ) for the difference of the j th intrinsic volume of K and the expectation of the j th intrinsic volume of K ( n ). We extend this formula to the case when the only condition on K is that a ball rolls freely inside K .
ISSN:0031-5303
1588-2829
DOI:10.1007/s10998-008-8143-4