Journal of Convex Analysis 33 (2026), No. 3&4, 1061--1076
Copyright Heldermann Verlag 2026
On Convex Cones for which Criticality Coincides with Antipodality
Alberto Seeger
Département de Mathématiques, Université d'Avignon, Avignon, France
aseegerfrance@gmail.com
Mounir Torki
LMA, Université d'Avignon, Avignon, France
mounir.torki@univ-avignon.fr
[Abstract-pdf]
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.
Keywords: Convex cone, maximal angle, critical angle, antipodal pair, opening angle function, diametric completeness, antipodal mate property.
MSC: 52A20, 52A40.
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