Characterizations of ellipsoids by means of the strong intersection property
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909608595947520 |
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| author | Morales-Amaya, E. |
| author_facet | Morales-Amaya, E. |
| contents | Let $E_1,E_2\subset \mathbb{R}^n$ be two homothetic solid ellipsoids, $n\geq 3$, with center at the origin $O$ of a system coordinates of $\mathbb{R}^n$, and $E_1\subset E_2$. Then there exists a $O$-symmetric ellipsoid $E_3$ such that $E_3$ is homothetic to $E_1$ and, for all $x\in \partial E_2$, there exists an hyperplano $Π(x)$, $O\in Π(x)$, such that the relation \begin{eqnarray} S(E_1,x)\cap S(E_1,-x)= Π(x) \cap E_3. \end{eqnarray} holds, where $S(E_1,x)$ and $S(E_1,-x)$ are the supporting cones of $E_1$ with apex $x$ and $-x$, respectively.
In this work we prove that aforesaid condition characterizes the ellipsoid. In fact, we prove that if $K,S, G\subset \mathbb{R}^n$ are three convex bodies, $n\geq 3$, $O\in K$, $K\subset G\subset S$ and $G$ strictly convex and, for all $x\in \partial S$, there exists $y\in \partial S$, $O$ in the line defined by $x,y$, an hyperplane $Π(x)$, $O\in Π(x)$, such that the relation \begin{eqnarray} S(K,x)\cap S(K,y)= Π(x) \cap \partial G. \end{eqnarray} holds, where $S(K,x)$ and $S(K,y)$ are the supporting cones of $K$ with apex $x$ and $y$, respectively, then $G,K$ and $S$ are $O$-symmetric homothetic ellipsoids.
In this case, we say that the convex body $K$ has the 'strong intersection property' relative to $O$ and $S$ and with 'associated' body $G$. Thus our main result affirm that if the convex body $K$ has the strong intersection property relative to $O$ and $S$ and with associated strictly convex body $G$, then $K,S$ and $G$ are concentric homothetic ellipsoids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_08169 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterizations of ellipsoids by means of the strong intersection property Morales-Amaya, E. Metric Geometry Let $E_1,E_2\subset \mathbb{R}^n$ be two homothetic solid ellipsoids, $n\geq 3$, with center at the origin $O$ of a system coordinates of $\mathbb{R}^n$, and $E_1\subset E_2$. Then there exists a $O$-symmetric ellipsoid $E_3$ such that $E_3$ is homothetic to $E_1$ and, for all $x\in \partial E_2$, there exists an hyperplano $Π(x)$, $O\in Π(x)$, such that the relation \begin{eqnarray} S(E_1,x)\cap S(E_1,-x)= Π(x) \cap E_3. \end{eqnarray} holds, where $S(E_1,x)$ and $S(E_1,-x)$ are the supporting cones of $E_1$ with apex $x$ and $-x$, respectively. In this work we prove that aforesaid condition characterizes the ellipsoid. In fact, we prove that if $K,S, G\subset \mathbb{R}^n$ are three convex bodies, $n\geq 3$, $O\in K$, $K\subset G\subset S$ and $G$ strictly convex and, for all $x\in \partial S$, there exists $y\in \partial S$, $O$ in the line defined by $x,y$, an hyperplane $Π(x)$, $O\in Π(x)$, such that the relation \begin{eqnarray} S(K,x)\cap S(K,y)= Π(x) \cap \partial G. \end{eqnarray} holds, where $S(K,x)$ and $S(K,y)$ are the supporting cones of $K$ with apex $x$ and $y$, respectively, then $G,K$ and $S$ are $O$-symmetric homothetic ellipsoids. In this case, we say that the convex body $K$ has the 'strong intersection property' relative to $O$ and $S$ and with 'associated' body $G$. Thus our main result affirm that if the convex body $K$ has the strong intersection property relative to $O$ and $S$ and with associated strictly convex body $G$, then $K,S$ and $G$ are concentric homothetic ellipsoids. |
| title | Characterizations of ellipsoids by means of the strong intersection property |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2505.08169 |