On convex bodies with constant non-central sections
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913079902601216 |
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| author | Haddad, J. Ryabogin, D. |
| author_facet | Haddad, J. Ryabogin, D. |
| contents | 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 $\frac 1π \arctan((\frac{3A}{4π})^{1/3})$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_00299 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On convex bodies with constant non-central sections Haddad, J. Ryabogin, D. Metric Geometry Number Theory 52A20, 52A38, 44A12, 11J70 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 $\frac 1π \arctan((\frac{3A}{4π})^{1/3})$ 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. |
| title | On convex bodies with constant non-central sections |
| topic | Metric Geometry Number Theory 52A20, 52A38, 44A12, 11J70 |
| url | https://arxiv.org/abs/2605.00299 |