On Convex Cones for which Criticality Coincides with Antipodality

Let

be a proper cone in a Euclidean space

E E

. An antipodal pair of

K K

is a pair of unit vectors in

K K

that achieves the maximum angle

\theta_{\rm max}(K) θ m a x ( K )

of the cone. A critical pair of

K K

is a pair of unit vectors in

K K

that satisfies the Karush-Kuhn-Tucker optimality conditions of the nonconvex optimization problem defining

\theta_{\rm max}(K) θ m a x ( K )

. Antipodality implies criticality, but not conversely. This work studies the class of proper cones for which antipodality and criticality coincide.