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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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