Characterization of the sphere by means of congruent support cones
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866910006395273216 |
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| author | Amaya, Efren Morales |
| author_facet | Amaya, Efren Morales |
| contents | Let $M$ be a convex body and let $K$ be a closed convex surface $K$ both contained in the Euclidean space $\mathbb{E}^3$. What can we say about $M$ if $K$ encloses $M$ and if from all the points in $K$ the body $M$ looks the same? In this work we are going to present a result which claims that if for every two support cones $C_x$, $C_y$ of $M$, with apexes $x,y \in K$, respectively, there exists $Φ$ in the semi direct product of the orthogonal group $O(3)$ and $\mathbb{E}^3$ such that $$C_y=Φ(C_x),$$ and this can be done in a continuous way, then $M$ is a sphere. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_20615 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterization of the sphere by means of congruent support cones Amaya, Efren Morales Metric Geometry Let $M$ be a convex body and let $K$ be a closed convex surface $K$ both contained in the Euclidean space $\mathbb{E}^3$. What can we say about $M$ if $K$ encloses $M$ and if from all the points in $K$ the body $M$ looks the same? In this work we are going to present a result which claims that if for every two support cones $C_x$, $C_y$ of $M$, with apexes $x,y \in K$, respectively, there exists $Φ$ in the semi direct product of the orthogonal group $O(3)$ and $\mathbb{E}^3$ such that $$C_y=Φ(C_x),$$ and this can be done in a continuous way, then $M$ is a sphere. |
| title | Characterization of the sphere by means of congruent support cones |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2502.20615 |