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On the approximation of a ball by random polytopes

K.-H. Küfer
1994 Advances in Applied Probability  
Let be a sequence of independent and identically distributed random vectors drawn from the d-dimensional unit ball Bd and let Xn be the random polytope generated as the convex hull of a 1,· ··, an. Furthermore, let Δ(Xn ): = Vol (Bd Xn ) be the volume of the part of the ball lying outside the random polytope. For uniformly distributed ai and 2 we prove that the limiting distribution of Δ(Xn )/Ε (Δ (Xn )) for n → ∞ (satisfies a 0–1 law. In particular, we show that Var for n → ∞. We provide
more » ... ∞. We provide analogous results for spherically symmetric distributions in Bd with regularly varying tail. In addition, we indicate similar results for the surface area and the number of facets of Xn.
doi:10.2307/1427895 fatcat:a6dm2vj7e5a4vhhzq75mu7tylu