Lexicographical characterization of the faces of convex sets (Q1384434)

Given a convex set \(C\subseteq\mathbb{R}^n\). A subset \(F\) is said to be a face of \(C\) if every closed line segment in \(C\) with a relative interior point \(F\) has both endpoints in \(F\). For instance, the set of points where a linear function attains its maximum over \(C\) is a face of \(C\). A face obtained in this way is called an exposed face. Not all faces of a convex set are necessarily exposed. As observed in \textit{R. T. Rockafellar} [`Convex analysis', Princeton University Press (1970; Zbl 0193.18401)], if \(C\) is the convex hull of a torus and \(D\) is one of the two closed disks forming the sides of \(C\), each relative boundary point of \(D\) is a face of \(C\) but not an exposed face; however, these points are exposed faces of \(D\), which is in turn an exposed face of \(C\). The aim of this paper is to show that all faces of a convex set are exposed in a lexicographical sense, namely, for each nonempty proper face \(F\) of a convex set \(C\) there exists a linear mapping the set of whose lexicographical maximum points is \(F\). As a consequence of this result, it turns out taking the exposed faces of a convex set \(C\), the exposed faces of the exposed faces, and so on, one obtains all the nonempty proper faces of \(C\).