  We give two characterizations of cones over ellipsoids in real normed vector
spaces. Let $C$ be a closed convex cone with nonempty interior such that $C$
has a bounded section of codimension $1$. We show that $C$ is a cone over an
ellipsoid if and only if every bounded section of $C$ has a center of symmetry.
We also show that $C$ is a cone over an ellipsoid if and only if the affine
span of $\partial C \cap \partial(a - C)$ has codimension $1$ for every point
$a$ in the interior of $C$. These results generalize the finite-dimensional
cases proved in (Jer\'onimo-Castro and McAllister, 2013).
