Conjugacy for closed convex sets
The authors generalize the definition of conjugate faces of a convex body. For a face \(F\) of a closed convex set \(C\) containing the origin they propose the following definition of the conjugate face \(F^{\triangle}\). Namely, \(F^{\triangle} = \{ y \in C^o \;| \;\forall_{x \in F} \;\langle x,y \rangle = -1 \}\) provided this set is non-empty, and otherwise \(F^{\triangle} = \{ y \in 0^+(C^o) \;| \;\forall_{x \in F} \;\langle x,y \rangle = -1 \}\). Here \(C^o\) means the polar set of \(C\) and \(0^+(C)\) denotes the recession cone of \(C\). A number of properties of the introduced conjugate faces is shown. For instance: if \(F\) is a face of \(C\) and if \(F^{\triangle} \not = \emptyset\), then \(F^{\triangle}\) is an exposed face of \(C^o\).
scientific article; zbMATH DE number 2153152