On strict inclusions in hierarchies of convex bodies

TL;DR

This paper investigates the hierarchy of convex bodies, demonstrating strict inclusions between classes of convex intersection bodies and their sections, revealing complex structural relationships in convex geometry.

Contribution

It establishes new strict inclusion results among classes of convex intersection bodies and their sections, advancing understanding of convex body hierarchies.

Findings

Proves that certain classes of convex intersection bodies are not contained within each other.

Shows that the hierarchy of convex bodies is strictly nested with no equalities.

Provides new insights into the structure of convex bodies and their sections.

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) $\mathcal I_l \not\subset \mathcal I_k$, $1\le k<l < n-3$.