Ellipsoidal cones in normed vector spaces

TL;DR

This paper characterizes cones over ellipsoids in real normed vector spaces using symmetry and boundary intersection properties, extending finite-dimensional results to infinite-dimensional settings.

Contribution

It provides two new characterizations of ellipsoidal cones in normed spaces, generalizing previous finite-dimensional theorems.

Findings

Cones over ellipsoids have symmetric bounded sections.

Boundary intersection conditions characterize ellipsoidal cones.

Results extend finite-dimensional characterizations to infinite-dimensional spaces.

Abstract

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).