Abstract

We discuss some aspects concerning the angular structure of a closed convex cone KK in a Euclidean vector space EE. The cone under consideration is assumed to be pointed and solid, but not necessarily smooth. Its Gauss map GKG_K is therefore to be understood in a multivalued sense. By definition, GKG_K assigns to a boundary point uu of KK the set GK(u):=NK(u)∩SEG_K(u):= N_K(u)\cap S_E, where SES_E is the unit sphere of EE and NKN_K is the normal cone map of KK in the sense of convex analysis. By a positive homogeneity argument, there is no loss of generality in assuming that uu has unit length. Among other issues, we elaborate on the connection between GK(u)G_K(u) and the set MK(u)M_K(u) of antipodal mates of uu. That vv is an antipodal mate of uu means that {u,v}\{u,v\} is a pair of unit vectors in the boundary of KK achieving the maximum angle of the cone.

Author information

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Keywords

  • Convex cone
  • maximum angle
  • largest angle function
  • oriented distance function
  • Gauss map
  • antipodal mate
  • diametric completeness

Mathematics Subject Classification

52A20, 52A40

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Suggested citation

A. Seeger, M. Torki. “The Gauss Map of a Nonsmooth Convex Cone and the Antipodal Mate Property.” Journal of Convex Analysis 32 (2025), No. 3, 835–850. https://doi.org/10.68381/jca32042

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Published by Heldermann Verlag, 2025. Rights now held by Banach Press LLC.