Abstract

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

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Keywords

  • Convex cone
  • maximal angle
  • critical angle
  • antipodal pair
  • opening angle function
  • diametric completeness
  • antipodal mate property

Mathematics Subject Classification

52A20, 52A40

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Suggested citation

A. Seeger, M. Torki. “On Convex Cones for which Criticality Coincides with Antipodality.” Journal of Convex Analysis 33 (2026), No. 3&4, 1061–1076. https://doi.org/10.68381/jca33056

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Published by Heldermann Verlag, 2026. Rights now held by Banach Press LLC.