Abstract view
The Mean Width of Circumscribed Random Polytopes


Published:20100726
Printed: Dec 2010
Károly J. Böröczky,
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary
Rolf Schneider,
Mathematisches Institut, AlbertLudwigsUniversität, Freiburg i. Br., Germany
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.