On strict inclusions in hierarchies of convex bodies

Abstract

Let I k \mathcal I_k be the class of convex k k -intersection bodies in R n \mathbb {R}^n (in the sense of Koldobsky) and I k m \mathcal I_k^m be the class of convex origin-symmetric bodies all of whose m m -dimensional central sections are k k -intersection bodies. We show that 1) I k m ⊄ I k m + 1 \mathcal I_k^m\not \subset \mathcal I_k^{m+1} , k + 3 ≤ m > n k+3\le m>n , and 2) I l ⊄ I k \mathcal I_l \not \subset \mathcal I_k , 1 ≤ k > l > n − 3 1\le k>l > n-3 .