Convex bodies with sections with hyperplanes of symmetry
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912598828515328 |
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| author | Morales-Amaya, Efrén |
| author_facet | Morales-Amaya, Efrén |
| contents | Let $K\subset \mathbb{R}^n$ be a convex body and let $p$ in the interior of $ K$, $n \geq 3$. The
point $p$ is said to be a \textit{Larman point} of $K$ if, for every hyperplane
$Π$ passing through $p$, the section $Π\cap K$ has a $(n-2)$-plane of
symmetry. If, in addition, for every hyperplane $Π$ passing through $p$, the
section $Π\cap K$ has a $(n-2)$-plane of symmetry which contains $p$,
then the point $p$ is called a revolution point. In this work we prove that if
for the convex body $K$, $n \geq 3$, there exists a
hyperplane $H$, a point $p$ such that $p$ is a Larman point of $K$ but
not a revolution point and, for every hyperplane $Π$ passing though $p$,
the section $Π\cap K$ has an $(n-2)$-plane of symmetry parallel to $H$,
then $K$ is an ellipsoid of revolution with an axis perpendicular to $H$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17326 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convex bodies with sections with hyperplanes of symmetry Morales-Amaya, Efrén Metric Geometry Let $K\subset \mathbb{R}^n$ be a convex body and let $p$ in the interior of $ K$, $n \geq 3$. The point $p$ is said to be a \textit{Larman point} of $K$ if, for every hyperplane $Π$ passing through $p$, the section $Π\cap K$ has a $(n-2)$-plane of symmetry. If, in addition, for every hyperplane $Π$ passing through $p$, the section $Π\cap K$ has a $(n-2)$-plane of symmetry which contains $p$, then the point $p$ is called a revolution point. In this work we prove that if for the convex body $K$, $n \geq 3$, there exists a hyperplane $H$, a point $p$ such that $p$ is a Larman point of $K$ but not a revolution point and, for every hyperplane $Π$ passing though $p$, the section $Π\cap K$ has an $(n-2)$-plane of symmetry parallel to $H$, then $K$ is an ellipsoid of revolution with an axis perpendicular to $H$. |
| title | Convex bodies with sections with hyperplanes of symmetry |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2509.17326 |