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Random polytopes in smooth convex bodies

  • Imre Bárány (a1)
  • DOI: http://dx.doi.org/10.1112/S0025579300006872
  • Published online: 01 February 2010
Abstract
Abstract.

Let K ⊂ Rd be a convex body and choose points xl, x2, …, xn randomly, independently, and uniformly from K. Then Kn = conv {x1, …, xn} is a random polytope that approximates K (as n → ∞) with high probability. Answering a question of Rolf Schneider we determine, up to first order precision, the expectation of vol K – vol Kn when K is a smooth convex body. Moreover, this result is extended to quermassintegrals (instead of volume).

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3.I. Bárány . Intrinsic volumes and f-vectors of random polytopes. Math. Ann., 285 (1989), 671699.

5.I. Bonnesen and W. Fenchel , Theorie der konvexen Körper (Springer, Berlin, 1934).

7.B. Efron . The convex hull of a random set of points. Biometrika, 52 (1965), 331343.

8.A. Rényi and R. Sulanke . Über die konvexe Hiille von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitsth verw. Geb., 2 (1963), 7584.

9.A. Rényi and R. Sulanke . Über die konvexe Hülle von n zufällig gewählten Punkten II. Z. Wahrscheinlichkeitsth. verw. Geb., 3 (1964), 138147.

11.R. Schneider . Random approximation of convex sets. J. Microscopy, 151 (1988), 211227.

12.R. Schneider and J. A. Wieacker . Random polytopes in a convex body. Z. Walrscheinlichkeitsth. verw. Geb., 52 (1980), 6973.

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Mathematika
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  • EISSN: 2041-7942
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