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An equichordal characterization of the ellipsoid and the sphere

Victor A. Aguilar-Arteaga, Rafael Iván Ayala-Figueroa, Jesús Jerónimo-Castro, Efrén Morales-Amaya

Abstract

Let $K$ and $L$ be two convex bodies in $\mathbb R^n$, $n\geq 3$, with $L\subset \text{int}\, K$. In this paper we prove the following result: if every two parallel chords of $K$, supporting $L$ have the same length, then $K$ and $L$ are homothetic and concentric ellipsoids. We also prove a similar theorem when instead of parallel chords we consider concurrent chords. We may also replace, in both theorems, supporting chords of $L$ by supporting sections of constant width. In the last section we also prove similar theorems where we consider projections instead of sections.

An equichordal characterization of the ellipsoid and the sphere

Abstract

Let and be two convex bodies in , , with . In this paper we prove the following result: if every two parallel chords of , supporting have the same length, then and are homothetic and concentric ellipsoids. We also prove a similar theorem when instead of parallel chords we consider concurrent chords. We may also replace, in both theorems, supporting chords of by supporting sections of constant width. In the last section we also prove similar theorems where we consider projections instead of sections.
Paper Structure (7 sections, 9 theorems, 2 equations, 5 figures)

This paper contains 7 sections, 9 theorems, 2 equations, 5 figures.

Table of Contents

  1. Introduction
  2. A characterization of the ellipsoid
  3. Characterization of the sphere by concurrent chords
  4. Characterization of the sphere by sections of constant width
  5. A particular case of Süss theorem
  6. Equichordal projections
  7. Statements and Declarations

Key Result

Theorem 1

Let $K,L \subset \mathbb{R}^{n}$, $n\geq 3$, be convex bodies with $L \subset \emph{int} K$. Suppose that for every $u \in \mathbb{S}^{n-1}$ all the chords of $K$ supporting $L$ and parallel to $u$, have the same length $\lambda (u)$. Then $K$ and $L$ are homothetic and concentric ellipsoids.

Figures (5)

  • Figure 1: Parallel chords tangent to $\mathcal{E}'$ have the same length
  • Figure 2: $[a,a']$ is the tangent chord with minimum length
  • Figure 3: The length of $[y,x]$ is bigger than the length of $[y',x']$
  • Figure 4: The shadow boundaries orthogonal to $\upsilon$ are planar curves
  • Figure 5: The section $S(a,b)$ is a set of constant width $1$

Theorems & Definitions (16)

  • Theorem 1
  • Conjecture 1
  • Conjecture 2
  • Lemma 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Theorem 4
  • Lemma 2
  • Theorem 5
  • ...and 6 more