A note on k-intersection bodies

The concept of an intersection body is central for the dual Brunn-Minkowski theory and has also played an important role in the solution of the Busemann-Petty problem. A more general concept of k-intersection bodies is related to the generalization of the Busemann-Petty problem. In this note, we compare classes of k-intersection bodies for different k and examine the conjecture that these classes increase with k. In particular, we construct a 4-intersection body that is not a 2-intersection body.