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The Mean Width of Circumscribed Random Polytopes |
Published:2010-07-26
Printed: Dec 2010
Printed: Dec 2010
- Károly J. Böröczky,
- Rolf Schneider,
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 |
| MSC Classifications: |
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 |