A Characterization of the sphere and a body of revolution by means of Larman points
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866911411962118144 |
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| author | Alfonseca, María Angeles Cordier, Michelle Jerónimo-Castro, Jesús Morales-Amaya, Efrén |
| author_facet | Alfonseca, María Angeles Cordier, Michelle Jerónimo-Castro, Jesús Morales-Amaya, Efrén |
| contents | Let $K\subset \mathbb{R}^n$, $n\geq 3$, be a convex body. A point $p$ the interior of $K$ is said to be a 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 $p$ is a Larman point of $K$ and, in addition, for every section $Π\cap K$, $p$ is in the corresponding $(n-2)$-plane of symmetry, then we call $p$ a revolution point of $K$. We conjecture that if $K$ contains a Larman point which is not a revolution point, then $K$ is either an ellipsoid or a body of revolution. This generalizes a conjecture of K. Bezdek for convex bodies in $\mathbb{R}^3$ to $n \geq 4$. We prove several results related to the conjecture for strictly convex origin symmetric bodies. Namely, if $K \subset \mathbb{R}^n$ is a strictly convex origin symmetric body that contains a revolution point $p$ which is not the origin, then $K$ is a body of revolution. This generalizes the False Axis of Revolution Theorem. We also show that if $p$ is a Larman point of $K \subset \mathbb{R}^3$ and there exists a line $L$ such that $p\notin L$ and, for every plane $Π$ passing through $p$, the line of symmetry of the section $Π\cap K$ intersects $L$, then $K$ is a body of revolution (in some cases, we conclude that $K$ is a sphere). We obtain a similar result for projections of $K$. Additionally, for $K \subset \mathbb{R}^n$, $n \geq 4$, we show that if every hyperplane section or projection of $K$ is a body of revolution and $K$ has a unique diameter $D$, then $K$ is a body of revolution with axis $D$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_09585 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Characterization of the sphere and a body of revolution by means of Larman points Alfonseca, María Angeles Cordier, Michelle Jerónimo-Castro, Jesús Morales-Amaya, Efrén Metric Geometry Let $K\subset \mathbb{R}^n$, $n\geq 3$, be a convex body. A point $p$ the interior of $K$ is said to be a 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 $p$ is a Larman point of $K$ and, in addition, for every section $Π\cap K$, $p$ is in the corresponding $(n-2)$-plane of symmetry, then we call $p$ a revolution point of $K$. We conjecture that if $K$ contains a Larman point which is not a revolution point, then $K$ is either an ellipsoid or a body of revolution. This generalizes a conjecture of K. Bezdek for convex bodies in $\mathbb{R}^3$ to $n \geq 4$. We prove several results related to the conjecture for strictly convex origin symmetric bodies. Namely, if $K \subset \mathbb{R}^n$ is a strictly convex origin symmetric body that contains a revolution point $p$ which is not the origin, then $K$ is a body of revolution. This generalizes the False Axis of Revolution Theorem. We also show that if $p$ is a Larman point of $K \subset \mathbb{R}^3$ and there exists a line $L$ such that $p\notin L$ and, for every plane $Π$ passing through $p$, the line of symmetry of the section $Π\cap K$ intersects $L$, then $K$ is a body of revolution (in some cases, we conclude that $K$ is a sphere). We obtain a similar result for projections of $K$. Additionally, for $K \subset \mathbb{R}^n$, $n \geq 4$, we show that if every hyperplane section or projection of $K$ is a body of revolution and $K$ has a unique diameter $D$, then $K$ is a body of revolution with axis $D$. |
| title | A Characterization of the sphere and a body of revolution by means of Larman points |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2307.09585 |