An equichordal characterization of the ellipsoid and the sphere
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917286937362432 |
|---|---|
| author | Aguilar-Arteaga, Victor A. Ayala-Figueroa, Rafael Iván Jerónimo-Castro, Jesús Morales-Amaya, Efrén |
| author_facet | Aguilar-Arteaga, Victor A. Ayala-Figueroa, Rafael Iván Jerónimo-Castro, Jesús Morales-Amaya, Efrén |
| contents | Let $K$ and $L$ be two convex bodies in $\mathbb R^n$, $n\geq 3$, with $L\subset \text{int}\, K$. In this paper we prove the following result: if every two parallel chords of $K$, supporting $L$ have the same length, then $K$ and $L$ are homothetic and concentric ellipsoids. We also prove a similar theorem when instead of parallel chords we consider concurrent chords. We may also replace, in both theorems, supporting chords of $L$ by supporting sections of constant width. In the last section we also prove similar theorems where we consider projections instead of sections. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_02956 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An equichordal characterization of the ellipsoid and the sphere Aguilar-Arteaga, Victor A. Ayala-Figueroa, Rafael Iván Jerónimo-Castro, Jesús Morales-Amaya, Efrén Metric Geometry Let $K$ and $L$ be two convex bodies in $\mathbb R^n$, $n\geq 3$, with $L\subset \text{int}\, K$. In this paper we prove the following result: if every two parallel chords of $K$, supporting $L$ have the same length, then $K$ and $L$ are homothetic and concentric ellipsoids. We also prove a similar theorem when instead of parallel chords we consider concurrent chords. We may also replace, in both theorems, supporting chords of $L$ by supporting sections of constant width. In the last section we also prove similar theorems where we consider projections instead of sections. |
| title | An equichordal characterization of the ellipsoid and the sphere |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2308.02956 |