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Characterizations of ellipsoids by means of the strong intersection property

E. Morales-Amaya

TL;DR

The paper develops a framework to characterize ellipsoids via a strong intersection property of support cones in ${\mathbb{R}}^n$. It defines the property relative to a point ${O}$ and an ambient body ${S}$ with an associated strictly convex body ${G}$, and proves that any ${K}$ satisfying this property with ${G}$ must lie in a concentric family of homothetic ellipsoids with ${S}$ and ${G}$; i.e., ${K},{S},{G}$ are $O$-symmetric concentric homothetic ellipsoids. The authors establish key results in dimension ${3}$ and then reduce higher dimensions to the 3D case via hyperplane sections, employing affine symmetry arguments and Kakutani's ellipsoid criterion. This work extends prior ellipsoid intersection results by providing a complete characterization under the strong intersection property within this convex-geometric framework.

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 $Π(x)$, $O\in Π(x)$, 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.

Characterizations of ellipsoids by means of the strong intersection property

TL;DR

The paper develops a framework to characterize ellipsoids via a strong intersection property of support cones in . It defines the property relative to a point and an ambient body with an associated strictly convex body , and proves that any satisfying this property with must lie in a concentric family of homothetic ellipsoids with and ; i.e., are -symmetric concentric homothetic ellipsoids. The authors establish key results in dimension and then reduce higher dimensions to the 3D case via hyperplane sections, employing affine symmetry arguments and Kakutani's ellipsoid criterion. This work extends prior ellipsoid intersection results by providing a complete characterization under the strong intersection property within this convex-geometric framework.

Abstract

Let be two homothetic solid ellipsoids, , with center at the origin of a system coordinates of , and . Then there exists a -symmetric ellipsoid such that is homothetic to and, for all , there exists an hyperplano , , such that the relation \begin{eqnarray} S(E_1,x)\cap S(E_1,-x)= Π(x) \cap E_3. \end{eqnarray} holds, where and are the supporting cones of with apex and , respectively. In this work we prove that aforesaid condition characterizes the ellipsoid. In fact, we prove that if are three convex bodies, , , and strictly convex and, for all , there exists , in the line defined by , an hyperplane , , such that the relation \begin{eqnarray} S(K,x)\cap S(K,y)= Π(x) \cap \partial G. \end{eqnarray} holds, where and are the supporting cones of with apex and , respectively, then and are -symmetric homothetic ellipsoids. In this case, we say that the convex body has the 'strong intersection property' relative to and and with 'associated' body . Thus our main result affirm that if the convex body has the strong intersection property relative to and and with associated strictly convex body , then and are concentric homothetic ellipsoids.
Paper Structure (7 sections, 12 theorems, 28 equations, 2 figures)

This paper contains 7 sections, 12 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction.
  2. Statement of the results.
  3. Proof of Theorems \ref{['tokio']} and \ref{['emperador']}.
  4. Proof of Theorem \ref{['great']} for dimension 3.
  5. Proof of Theorem \ref{['chiquita']} for dimension 3.
  6. Proof of Theorem \ref{['valor']} for dimension 3.
  7. Reduction of the general case of Theorems \ref{['valor']}, \ref{['great']} and \ref{['chiquita']} to dimension 3.

Key Result

Theorem 1

Let $E\subset \mathbb{R}^{n}$ be an $O$-symmetric ellipsoid, $n\geq 3$, and let $S\subset \mathbb{R}^{n}$ be an embedding of $\mathbb{S}^{n-1}$ in $\mathbb{R}^{n}$ such that $S$ is $O$-star and $E\subset \operatorname*{int} S$. Then for all $x\in S$ there exists an hyperplane $\Pi(x)$, $O\in \Pi(x)$ holds.

Figures (2)

  • Figure 1: The relation $R^{\Lambda}_{pq}(\Sigma(K,p))=\Sigma(K,q)$ holds.
  • Figure 2: Given the plane $\Lambda$, $O\in \Lambda$, there exists $p\in \operatorname*{bd} S$ such that $C(K,p)\cap C(K,-p)=\Lambda \cap G$.

Theorems & Definitions (20)

  • Theorem 1
  • Theorem 2
  • Conjecture 1
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 10 more