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Expectation of random polytopes. (English) Zbl 0873.52006

Let \(C\) be a convex body (with interior points) in Euclidean \(d\)-space and \(P_n\) the convex hull of \(n\) independent, identically distributed random points in \(C\). The set-valued expectation \(E_n\) is then a convex body in \(C\). Under suitable conditions on the distribution on the random points, \(E_n\) tend to \(C\) (as \(n\to\infty\)) in the Hausdorff metric. Here, the author obtains a number of interesting results on the exact asymptotic behaviour of the deviation of the support functions \(h_C-h_{E_n}\) in the following cases:
(i) \(C\) is of class \({\mathcal C}^k\), \(k\geq 3\), with positive Gauss curvature and the distribution is uniform on \(C\) (and a more specific result for \(d=2)\),
(ii) \(C\) is of class \({\mathcal C}^2\) with positive Gauss curvature and the distribution has a continuous density (with respect to the Lebesgue measure),
(iii), (iv) the same as (i), (ii) with random points on the boundary of \(C\).
He also shows that for most convex bodies \(C\) (in the sense of Baire category) and uniformly distributed points on \(C\) (respectively on the boundary of \(C\)), the asymptotic behaviour of the Hausdorff distance \(\delta^H(C,E_n)\) is extremely irregular.
Reviewer: W.Weil (Karlsruhe)

MSC:

52A22 Random convex sets and integral geometry (aspects of convex geometry)
60D05 Geometric probability and stochastic geometry
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
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