Isomorphic properties of intersection bodies

AbstractWe 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 byo.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, thendBM(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