Abstract
Let be a convex body in which slides freely in a ball. Let denote the intersection of closed half-spaces containing 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 and , and an asymptotic upper bound on the variance of the volume of . 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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