On Convex Cones for which Criticality Coincides with Antipodality

Let $K$ be a proper cone in a Euclidean space $E$. An antipodal pair of $K$ is a pair of unit vectors in $K$ that achieves the maximum angle $\theta_{\rm max}(K)$ of the cone. A critical pair of $K$ is a pair of unit vectors in $K$ that satisfies the Karush-Kuhn-Tucker optimality conditions of the nonconvex optimization problem defining $\theta_{\rm max}(K)$. Antipodality implies criticality, but not conversely. This work studies the class of proper cones for which antipodality and criticality coincide.