Abstract
Let K be a convex body in ℝd, 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 jth intrinsic volume of K and the expectation of the jth 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.
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Supported by OTKA grants 068398 and 049301, and by the EU Marie Curie TOK project DiscConvGeo.
Funded by the Marie-Curie Research Training Network “Phenomena in High-Dimensions” (MRTN-CT-2004-511953).
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Böröczky, K.J., Hoffmann, L.M. & Hug, D. Expectation of intrinsic volumes of random polytopes. Period Math Hung 57, 143–164 (2008). https://doi.org/10.1007/s10998-008-8143-4
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DOI: https://doi.org/10.1007/s10998-008-8143-4