On strict inclusions in hierarchies of convex bodies

Yaskin, Vladyslav

doi:10.1090/S0002-9939-08-09424-0

On strict inclusions in hierarchies of convex bodies
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Proc. Amer. Math. Soc. 136 (2008), 3281-3291 Request permission

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

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