The mean width of random polytopes circumscribed around a convex body

摘要

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.