Polyhedral Subspaces of $L_{p}$ and Polars of Zonotopes

Abstract

Let $K$ be an origin-symmetric full-dimensional convex polytope in $\mathbb R^n$, $n\ge 3$. We show that, for $-n+3<p<1$, the normed space $(\mathbb R^n,\|\cdot\|_K)$ embeds in $L_p$ if and only if it embeds in $L_1$. The case of $p\le 0$ is understood in the generalized sense. In particular, an origin-symmetric full-dimensional convex polytope $K\subset\mathbb R^n$, $n\ge 5$, is an intersection body if and only if $K$ is the polar of a zonotope.

Disclosure

“H)◦H , so every hyperplane projection of K ◦ is a zonotope. Lemma 6 now implies that K ◦ is a zonotope. Thus, (1) implies (4), and the proof is complete. □ AI disclosure. Lemma 4 was obtained with the assistance of ChatGPT. ChatGPT was also used to proofread the manuscript and to improve its exposition and wording. References [1] E. D. Bolker, A class of convex bodies, Trans. Amer. Math. Soc. 145 (1969), 323–3”

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Verified

2026-08-23T05:20:35.849193Z

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