THE VOLUME OF RANDOM POLYTOPES CIRCUMSCRIBED AROUND A CONVEX BODY. Issue 1 (22nd June 2015)

• Record Type: Journal Article
• Authors: Fodor, Ferenc; Hug, Daniel; Ziebarth, Ines
• Is Part Of: Mathematika. Volume 62:Issue 1(2016)
• Page Start: 283
• Page End: 306
• Publication Date: 2015-06-22
• Subjects: 52A22 (primary) -- 60D05 -- 52A27 (secondary); Mathematics -- Periodicals; 510.5
• DOI: 10.1112/S0025579315000170
• Languages: English
• ISSNs: 0025-5793

Abstract

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.