On strict inclusions in hierarchies of convex bodies

Let ${\mathcal{I}}_k$ be the class of convex $k$-intersection bodies in ${\mathbb{R}}^n$ (in the sense of Koldobsky) and ${\mathcal{I}}_k^m$ be the class of convex origin-symmetric bodies all of whose $m$-dimensional central sections are $k$-intersection bodies. We show that 1) $\mathcal I_k^m\not\subset \mathcal I_k^{m+1}$,$k+3\le m<n$,and2)$\mathcal I_l \not\subset \mathcal I_k$,$1\le k<l < n-3$.