Let K be a convex body in R-d which slides freely in a ball. Let K-(n) denote the intersection of n closed half-spaces containing K whose bounding hyperplanes are independent and identically distributed according to a certain prescribed probability distribution. We prove an asymptotic formula for the expectation of the difference of the volumes of K-(n) and K, and an asymptotic upper bound on the variance of the volume of K-(n). We obtain these asymptotic formulas by proving results for weighted mean width approximations of convex bodies that admit a rolling ball by inscribed random polytopes and then using polar duality to convert them into statements about circumscribed random polytopes.
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