The mean width of circumscribed random polytopes

Abstract

For a given convex body K in $R^d$, 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.