Inscribing a regular octahedron in a three-dimensional convex body with smooth boundary (Q1781290)

The author solves a problem discussed earlier in [\textit{V. Klee} and \textit{S. Wagon}, Old and new unsolved problems in plane geometry and number theory (1991; Zbl 0784.51002, German translation 1997; Zbl 0882.00003), p. 140; \textit{T. Hausel, E. Makai} and \textit{A. Szücs}, Gen. Math. 5, 183--190 (1997; Zbl 0984.52003)]. Theorem. Let \(\|\cdot\|\) be a norm in the standard Euclidean space \(\mathbb R^3\). Let \(K\) be a compact convex body with smooth boudary \(\partial K\). Then \(\partial K\) contains the vertices \(A\), \(A'\), \(B\), \(B'\), \(C\), \(C'\) of a regular octahedron such that \(AA' = BB' \geq CC'\).