Characterizations of ellipsoids by means of the strong intersection property

TL;DR

This paper characterizes ellipsoids through a strong intersection property involving supporting cones and hyperplanes, proving that certain geometric conditions imply bodies are concentric homothetic ellipsoids.

Contribution

It introduces and proves a new characterization of ellipsoids based on the strong intersection property involving supporting cones and hyperplanes.

Findings

Supports the characterization of ellipsoids via the strong intersection property.

Shows that bodies satisfying the property are necessarily concentric homothetic ellipsoids.

Provides a geometric condition that uniquely identifies ellipsoids among convex bodies.

Abstract

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 $\Pi(x)$, $O\in \Pi(x)$, such that the relation \begin{eqnarray} S(E_1,x)\cap S(E_1,-x)= \Pi(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 $\Pi(x)$, $O\in \Pi(x)$, such that the relation \begin{eqnarray} S(K,x)\cap S(K,y)= \Pi(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.