Convex bodies with equipotential circles
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866912863245828096 |
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| author | González-García, Iván Jerónimo-Castro, Jesús Jiménez-Desantiago, Valentín Morales-Amaya, Efrén |
| author_facet | González-García, Iván Jerónimo-Castro, Jesús Jiménez-Desantiago, Valentín Morales-Amaya, Efrén |
| contents | Given a convex body $K\subset \mathbb R^2$ we say that a circle $Ω\subset \text{int} \ K$ is an equipotential circle if every tangent line of $Ω$ cuts a chord $AB$ in $K$ such that for the contact point $P=Ω\cap AB$ it holds that $|AP|\cdot|PB|=λ$, for a suitable constant number $λ$. The main result in this article is the following: Let $K\subset\mathbb R^2$ be a convex body which has an equipotential circle $\mathcal B$ with centre $O$ in its interior. Then $K$ has centre of symmetry at $O$, moreover, if none chord of $K$ which is tangent to $\mathcal B$ subtends an angle $π/2$ from $O$, then $K$ is a disc. We also derive some results which characterizes the ellipsoid and the sphere in $\mathbb R^3$ and introduce also the concept of equireciprocal disc. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2107_11670 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Convex bodies with equipotential circles González-García, Iván Jerónimo-Castro, Jesús Jiménez-Desantiago, Valentín Morales-Amaya, Efrén Metric Geometry Given a convex body $K\subset \mathbb R^2$ we say that a circle $Ω\subset \text{int} \ K$ is an equipotential circle if every tangent line of $Ω$ cuts a chord $AB$ in $K$ such that for the contact point $P=Ω\cap AB$ it holds that $|AP|\cdot|PB|=λ$, for a suitable constant number $λ$. The main result in this article is the following: Let $K\subset\mathbb R^2$ be a convex body which has an equipotential circle $\mathcal B$ with centre $O$ in its interior. Then $K$ has centre of symmetry at $O$, moreover, if none chord of $K$ which is tangent to $\mathcal B$ subtends an angle $π/2$ from $O$, then $K$ is a disc. We also derive some results which characterizes the ellipsoid and the sphere in $\mathbb R^3$ and introduce also the concept of equireciprocal disc. |
| title | Convex bodies with equipotential circles |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2107.11670 |