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The Mean Width of Circumscribed Random Polytopes

  Published:2010-07-26
 Printed: Dec 2010
  • Károly J. Böröczky,
    Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary
  • Rolf Schneider,
    Mathematisches Institut, Albert-Ludwigs-Universität, Freiburg i. Br., Germany
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Abstract

For a given convex body $K$ in ${\mathbb R}^d$, a random polytope $K^{(n)}$ is defined (essentially) as the intersection of $n$ independent closed halfspaces containing $K$ and having an isotropic and (in a specified sense) uniform distribution. We prove upper and lower bounds of optimal orders for the difference of the mean widths of $K^{(n)}$ and $K$ as $n$ tends to infinity. For a simplicial polytope $P$, a precise asymptotic formula for the difference of the mean widths of $P^{(n)}$ and $P$ is obtained.
Keywords: random polytope, mean width, approximation random polytope, mean width, approximation
MSC Classifications: 52A22, 60D05, 52A27 show english descriptions Random convex sets and integral geometry [See also 53C65, 60D05]
Geometric probability and stochastic geometry [See also 52A22, 53C65]
Approximation by convex sets
52A22 - Random convex sets and integral geometry [See also 53C65, 60D05]
60D05 - Geometric probability and stochastic geometry [See also 52A22, 53C65]
52A27 - Approximation by convex sets
 

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