Isomorphic properties of intersection bodies

Abstract

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 B P 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 B P k n can be estimated by o.v.r. ( K , B P k n ) : = i n f { ( | C | | K | ) 1 n : C ∈ B P k n , K ⊆ C } ≤ c n k l o g e n 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 I k ( K ) exists and is convex, then d B M ( I k ( K ) , B 2 n ) ≤ c ( k ) , where c ( k ) is a constant depending only on k, d B M 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.