  We give two characterizations of cones over ellipsoids. Let $C$ be a closed
pointed convex linear cone in a finite-dimensional real vector space. We show
that $C$ is a cone over an ellipsoid if and only if the affine span of
$\partial C \cap \partial(a - C)$ has dimension $\dim(C) - 1$ for every point
$a$ in the relative interior of $C$. We also show that $C$ is a cone over an
ellipsoid if and only if every bounded section of $C$ by an affine hyperplane
is centrally symmetric.
