On the variance of the mean width of random polytopes circumscribed around a convex body

Abstract

Let $$ be a convex body in $$ in which a ball rolls freely and which slides freely in a ball. Let $$ be the intersection of $$ i.i.d. random half-spaces containing $$ chosen according to a certain prescribed probability distribution. We prove an asymptotic upper bound on the variance of the mean width of $$ as $$. We achieve this result by first proving an asymptotic upper bound on the variance of the weighted volume of random polytopes generated by $$ i.i.d. random points selected according to certain probability distributions, then, using polarity, we transfer this to the circumscribed model.