Characterizations of ellipsoids by means of the strong intersection property

Let $E_1,E_2\subset {\mathbb{R}}^n$ be two homothetic solid ellipsoids, $n\ge 3$, 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 $x\in \mathrm{\partial }{E}_2$, there exists an hyperplano $\mathrm{\Pi }(x)$, $O\in \mathrm{\Pi }(x)$, 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, $n\ge 3$, $O\in K$, $K\subset G\subset S$ and $G$ strictly convex and, for all $x\in \mathrm{\partial }S$, there exists $y\in \mathrm{\partial }S$, $O$ in the line defined by $x,y$, an hyperplane $\mathrm{\Pi }(x)$, $O\in \mathrm{\Pi }(x)$, 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.