Only Solid Spheres Admit a False Axis of Revolution

Let $K\subset \mathbb{R}^{3}$ be a convex body. A point $p_{0}$ is a point of revolution for $K$ if every section of $K$ through $p_{0}$ has an axis of symmetry that passes through $p_{0}$. In particular, every point that lies in an axis of revolution is a point of revolution. A line $L\subset \mathbb{R}^3$ is a false axis of revolution, if every point of $L$ is a point of revolution for $K$ but $L$ is not an axis of revolution. The purpose of this paper is to prove that only solid spheres admit a false axis of revolution