Convex bodies with centrally symmetric sections
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917093250695168 |
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| author | Morales-Amaya, E. |
| author_facet | Morales-Amaya, E. |
| contents | Let $K\subset \mathbb{R}^n$ be a convex body, $n\geq 3$. We say that $K$ satisfies the Barker-Larman condition if there exists a ball $B$ in the interior of $K$ such that for every suppor hyperplane $Π$ of $B$, the section $Π\cap K$ is a centrally symmetric set.
Barker and Larman conjectured that the Barker-Larman condition characterizes the ellipsoid. In this work we prove an special case of such conjecture, in particular, we assume that the convex body $K$ is centrally symmetric. Our main result is the following: Let $K$ be a centrally symmetric and strictly convex body, with center at $O$, and let $B$ be a ball in the interior of $K$ and not containing $O$: If $K$ satisfies the Barker-Larman condition with respect to $B$ and $B$ is suitable for $K$ (intuitively, $B$ is suitable for $K$ if the boundary of $B$ is not very close to the boundary of $K$), then $K$ is an ellipsoid. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_07624 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Convex bodies with centrally symmetric sections Morales-Amaya, E. Metric Geometry Let $K\subset \mathbb{R}^n$ be a convex body, $n\geq 3$. We say that $K$ satisfies the Barker-Larman condition if there exists a ball $B$ in the interior of $K$ such that for every suppor hyperplane $Π$ of $B$, the section $Π\cap K$ is a centrally symmetric set. Barker and Larman conjectured that the Barker-Larman condition characterizes the ellipsoid. In this work we prove an special case of such conjecture, in particular, we assume that the convex body $K$ is centrally symmetric. Our main result is the following: Let $K$ be a centrally symmetric and strictly convex body, with center at $O$, and let $B$ be a ball in the interior of $K$ and not containing $O$: If $K$ satisfies the Barker-Larman condition with respect to $B$ and $B$ is suitable for $K$ (intuitively, $B$ is suitable for $K$ if the boundary of $B$ is not very close to the boundary of $K$), then $K$ is an ellipsoid. |
| title | Convex bodies with centrally symmetric sections |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2307.07624 |