On strict inclusions in hierarchies of convex bodies

Abstract

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) ${\mathcal{I}}_k^m\not\subset {\mathcal{I}}_k^{m+1}$, $k+3\le m<n$, and 2) ${\mathcal{I}}_l\not\subset {\mathcal{I}}_k$, $1\le k<l<n-3$.

10 pages