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Volumes of a random polytope in a convex set. (English) Zbl 0751.52003

Applied geometry and discrete mathematics, Festschr. 65th Birthday Victor Klee, DIMACS, Ser. Discret. Math. Theor. Comput. Sci. 4, 175-180 (1991).
[For the entire collection see Zbl 0726.00015.]
For fixed area of a compact convex set \(K\) in \(\mathbb{R}^ 2\) the maximal area of the convex hull of \(n\) i.i.d. uniformly distributed random points in \(K\) is obtained when \(K\) is a triangle. (W. Blaschke proved this for \(n=3\).)
Moreover, for higher dimensions \(d\) a similar result is shown for polytopes with at most \(d+2\) vertices.
Reviewer: U.Zaehle

MSC:

52A22 Random convex sets and integral geometry (aspects of convex geometry)
52A10 Convex sets in \(2\) dimensions (including convex curves)
52A38 Length, area, volume and convex sets (aspects of convex geometry)
60D05 Geometric probability and stochastic geometry

Citations:

Zbl 0726.00015
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