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arXiv · 2607.10881

The isomorphic section-projection problem for convex bodies

Johannes Hosle
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Abstract

Let $K, L$ be convex bodies in $\mathbb{R}^n$ with $K$ centered. Assume that $|K \cap \theta^{\perp}| \le |L|\theta^{\perp}|$ for all $\theta \in S^{n-1}$. We prove that $|K| \le c\sqrt{n}|L|$, which is sharp up to the choice of the absolute constant. The result gives the sharp isomorphic order in a mixed section-projection comparison problem, complementing the isomorphic Busemann-Petty and Shephard problems. It also removes the John's position assumption from an earlier result of the author, up to an absolute constant factor.

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Johannes Hosle. 2026-07-12. The isomorphic section-projection problem for convex bodies. https://arxiv.org/abs/2607.10881

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