Isomorphic properties of intersection bodies

We study isomorphic properties of two generalizations of intersection bodies – the class Ikn of k  -intersection bodies in RnRn and the class BPkn of generalized k  -intersection bodies in RnRn. 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 RnRn and 1≤k≤n−11≤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>0c>0 is an absolute constant. Next we prove that if K   is a symmetric convex body in RnRn, 1≤k≤n−11≤k≤n−1 and its k  -intersection body Ik(K)Ik(K) exists and is convex, thendBM(Ik(K),B2n)≤c(k), where c(k)c(k) is a constant depending only on k  , dBMdBM is the Banach–Mazur distance, and B2n is the unit Euclidean ball in RnRn. This generalizes a well-known result of Hensley and Borell. We conclude the paper with volumetric estimates for k-intersection bodies.