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| Év szerint | Hónap szerint | Ugrás a hónaphoz | |||||||
Papvári Dániel: Véletlen politópok súlyozott vegyes térfogatainak várható értéke |
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| Let K be a convex body in R^d, let j\in \{1,\ldots,d-1\} and let \varrho be a suitable probability density function with respect to the $d$-dimensional Hausdorff measure on K. Denote by $K_{(n)}$ the convex hull of $n$ points chosen randomly and independently from K according to the probability distribution determined by \varrho. For the case when \varrho\equiv 1/V(K) and \partial K is C^2_+, Reitzner (2004) 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. Böröczky, Hoffmann, and Hug (2008) extended this result to the case when \varrho\equiv 1/V(K) and the only condition on K is that a ball rolls freely in K. Böröczky, Fodor, Reitzner, and Vígh (2009) also showed that in general, the assumption of the existence of a rolling ball inside K, for the mean width, cannot be dropped. Böröczky, Fodor, and Hug (2010) proved an asymptotic formula for the weighted volume approximation of K under no smoothness assumptions on \partial K. We study the expectation of weighted intrinsic volumes for random polytopes generated by non-uniform probability distributions in convex bodies with very mild smoothness conditions. |
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