Let K and L be two convex bodies in \(\mathbb {R}^n\) , \(n\ge 3\) , 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.
