Some characterizations of ellipsoids by sections (Q1889919)

If \(K\subset\mathbb{R}^d\) \((d\geq 3)\) is a \(d\)-dimensional convex body and \(H\) is a hyperplane, then \(K\) is called \(H\)-homothetic (\(H\)-elliptic) if all \((d-1)\)-dimensional sections of \(K\) with hyperplanes parallel to \(H\) are homothetic (respectively, \((d- 1)\)-ellipsoids). Theorem 1: If there exist nonparallel hyperplanes \(H_1\), \(H_2\) such that \(K\) is \(H_i\)-homothetic and \(H_i\)-elliptic for \(i= 1,2\), then \(K\) is an ellipsoid. Examples show that this is not true with only one of the properties, `\(H_i\)-homothetic' or `\(H_i\)-elliptic'. Theorem 2: If \(K\) is centrally symmetric and there exist pairwise nonparallel hyperplanes \(H_1\), \(H_2\), \(H_3\) such that \(K\) is \(H_i\)-elliptic for \(i= 1,2,3\), then \(K\) is an ellipsoid. It is shown by nice examples that this is not true without the assumption of central symmetry.