An equichordal characterization of the ellipsoid and the sphere

TL;DR

This paper characterizes ellipsoids and spheres through chord length conditions, showing that specific equalities imply the bodies are homothetic, concentric ellipsoids, extending classical geometric characterizations.

Contribution

It introduces new characterizations of ellipsoids based on equal chord lengths, including parallel and concurrent chords, and extends to sections and projections.

Findings

Equal parallel chords imply bodies are homothetic ellipsoids.

Similar results hold for concurrent chords and sections of constant width.

Projections also characterize ellipsoids under certain conditions.

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.