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Random polytopes in smooth convex bodies

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Imre Bárány
Imre Bárány
Affiliation:
The Mathematical Institute of the Hungarian Academy of Sciences, 1364 Budapest P.O.B. 127, Hungary.
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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).

MSC classification

Secondary: 52A22: Random convex sets and integral geometry
Type
Research Article
Information
Mathematika , Volume 39 , Issue 1 , June 1992 , pp. 81 - 92
Copyright
Copyright © University College London 1992

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References

1.Affentranger, F.. The convex hull of random points with spherically symmetric distribution. To appear in Rend. Sem. Mat. Torino.Google Scholar
2.Affentranger, F. and Wieacker, J. A.. On the convex hull of uniform random points in a simple d-polytope. Discrete and Comp. Geometry, 6 (1991), 191–205.Google Scholar
3.Bárány, I.. Intrinsic volumes and f-vectors of random polytopes. Math. Ann., 285 (1989), 671–699.CrossRefGoogle Scholar
4.Bárány, I. and Buchta, C.. On the convex hull of uniform random points in an arbitrary d-polytope. Auz. Öster. Akad. Wiss. Matk-Natur., 77 (1990), 25–27.Google Scholar
5.Bonnesen, I. and Fenchel, W., Theorie der konvexen Körper (Springer, Berlin, 1934).Google Scholar
6.Bárány, I. and Larman, D. G.. Convex bodies, economic cap coverings, random polytopes. Mathematika, 35 (1988), 274–291.CrossRefGoogle Scholar
7.Efron, B.. The convex hull of a random set of points. Biometrika, 52 (1965), 331–343.CrossRefGoogle Scholar
8.Rényi, A. and Sulanke, R.. Über die konvexe Hiille von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitsth verw. Geb., 2 (1963), 75–84.CrossRefGoogle Scholar
9.Rényi, A. and Sulanke, R.. Über die konvexe Hülle von n zufällig gewählten Punkten II. Z. Wahrscheinlichkeitsth. verw. Geb., 3 (1964), 138–147.CrossRefGoogle Scholar
10.Santaló, L. A.. Integral Geometry and Geometric Probability (Addison-Wesley, Reading, MA, 1976).Google Scholar
11.Schneider, R.. Random approximation of convex sets. J. Microscopy, 151 (1988), 211–227.CrossRefGoogle Scholar
12.Schneider, R. and Wieacker, J. A.. Random polytopes in a convex body. Z. Walrscheinlichkeitsth. verw. Geb., 52 (1980), 69–73.CrossRefGoogle Scholar
13.Wieacker, J. A.. Einige Probleme der polyedrischen Approximation (Diplomarbeit, Freiburg i. Br., 1978).Google Scholar
14.Wei, B. F. van. The convex hull of a uniform sample from the interior of a simple d-polytope. J. Appl. Prob., 26 (1989), 259–273.Google Scholar