Isomorphic properties of intersection bodies

We study isomorphic properties of two generalizations of intersection bodies - the class I(k)(n) of k-intersection bodies in R(n) and the class BP(k)(n) of generalized k-intersection bodies in R(n). 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 R(n) and 1 <= k <= n - 1 then the outer volume ratio distance from K to the class BP(k)(n) can be estimated by o.v.r.(K, BP(k)(n)) : = inf{|C|/|K|)1/n : C is an element of BP(k)(n), K subset of C} <= c root n/k log en/k, where c > 0 is an absolute constant. Next we prove that if K is a symmetric convex body in R(n), 1 <= k <= n - 1 and its k-intersection body 1(k) (K) exists and is convex, then d(Bm) (I(k)(K), B(2)(n)) <= c(k), where c(k) is a constant depending only on k, d(BM) is the Banach-Mazur distance, and B(2)(n) is the unit Euclidean ball in R(n). This generalizes a well-known result of Hensley and Borell. We conclude the paper with volumetric estimates for k-intersection bodies. (C) 2011 Elsevier Inc. All rights reserved.