Isomorphic properties of intersection bodies Academic Article uri icon

abstract

  • We study isomorphic properties of two generalizations of intersection bodies - the class Ikn of k-intersection bodies in Rn and the class BPkn of generalized k-intersection bodies in Rn. In particular, we show that all convex bodies can be in a certain sense approximated by intersection bodies, namely, if K is any symmetric convex body in Rn and 1≤k≤n-1 then the outer volume ratio distance from K to the class BPkn can be estimated by. o.v.r.(K,BPkn):=inf{(|C||K|)1n:C∈BPkn,K≤C}≤cnklogenk, where c>0 is an absolute constant. Next we prove that if K is a symmetric convex body in Rn, 1≤k≤n-1 and its k-intersection body Ik(K) exists and is convex, then. dBM(Ik(K),B2n)≤c(k), where c(k) is a constant depending only on k, dBM is the Banach-Mazur distance, and B2n is the unit Euclidean ball in Rn. This generalizes a well-known result of Hensley and Borell. We conclude the paper with volumetric estimates for k-intersection bodies. © 2011 Elsevier Inc.

author list (cited authors)

  • Koldobsky, A., Paouris, G., & Zymonopoulou, M.

citation count

  • 16

publication date

  • November 2011