THE VOLUME OF RANDOM POLYTOPES CIRCUMSCRIBED AROUND A CONVEX BODY. Issue 1 (22nd June 2015) - Fodor, Ferenc; Hug, Daniel; Ziebarth, Ines
Abstract : 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.