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arXiv · 2308.02956

An equichordal characterization of the ellipsoid and the sphere

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

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Victor A. Aguilar-Arteaga, Rafael Iván Ayala-Figueroa, Jesús Jerónimo-Castro, Efrén Morales-Amaya. 2023-08-05. An equichordal characterization of the ellipsoid and the sphere. https://arxiv.org/abs/2308.02956

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