The mean width of random polytopes circumscribed around a convex body

Let K be a d-dimensional convex body, and let K 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 and K, and another asymptotic formula for the expectation of the number of facets of K. These results are achieved by establishing an asymptotic result on weighted volume approximation of K and by “dualizing” it using polarity.