The Gauss Map of a Nonsmooth Convex Cone and the Antipodal Mate Property

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