The mean width of random polytopes circumscribed around a convex body

Abstract

Let K be a d-dimensional convex body, and let $K^{(n)}$ be the intersection of n halfspaces containing $K$ whose bounding hyperplanes are independent and identically distributed. Under suitable distributional assumptions, we prove an asymptotic formula for the expectation of the difference of the mean widths of $K^{(n)}$ and K, and another asymptotic formula for the expectation of the number of facets of $K^{(n)}$. These results are achieved by establishing an asymptotic result on weighted volume approximation of $K$ and by "dualizing" it using polarity.