Abstract :
We study isomorphic properties of two generalizations of intersection bodies – the class In
k of kintersection
bodies in Rn and the class BPn
k 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 BPn
k can be estimated by
o.v.r. K,BPn
k := inf |C|
|K|
1
n
: C ∈ BPn
k, K ⊆ C c n
k
log
en
k
,
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),Bn
2 c(k),
where c(k) is a constant depending only on k, dBM is the Banach–Mazur distance, and Bn
2 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. All rights reserved
