The Mean Width of Circumscribed Random Polytopes

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.

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