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arXiv · 2605.00299

On convex bodies with constant non-central sections

J. Haddad·D. Ryabogin
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Abstract

We prove that if $C$ is a symmetric convex body of revolution in $\mathbb R^4$ containing the unit Euclidean ball $\mathbb B_4$, such that the sections of $C$ by hyperplanes tangent to $\mathbb B_4$ have constant area $A>0$, then $C$ is a Euclidean ball, provided $\frac 1{\pi} \arctan((\frac{3A}{4\pi})^{1/3})$ satisfies certain arithmetic properties that can be read from its expansion as a continued fraction. We show that the set of values $A$ satisfying these properties has positive Hausdorff dimension.

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J. Haddad, D. Ryabogin. 2026-05-01. On convex bodies with constant non-central sections. https://arxiv.org/abs/2605.00299

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