A note on touching cones and faces (Q2903480)

In this work, the author considers convex sets in a finite-dimensional Euclidean vector space \(\mathbb E\) and studies relationships between their faces, exposed faces, normal cones and touching cones. This is done in the framework of lattice theory, the four lattices naturally defined for each set of these objects. For a given convex set \(C \subset \mathbb E,\) a convex subset \(F \subset C\) is called a face if for all \(x, y \in C\) nonempty intersection of \(F\) with an open segment \(]x, y[\) implies that the closed segment \([x, y] \subset F.\) The set of all faces, \(\mathcal F (C),\) including also the improper faces \(\emptyset\) and \(C\), forms a lattice -- \textit{the face lattice} of \(C.\) An exposed face of \(C\) is the intersection of \(C\) with a supporting hyperplane. The set of all exposed faces, \(\mathcal F_{\perp}(C),\) is \textit{the exposed face lattice}. The normal cone of \(C\) at \(x \in C\) is defined by \( N (C, x):= \{ u \in \mathbb E : \langle u , y - x \rangle\;\leq 0 , \; \forall y \in C\}\) and for a nonempty convex subset \(F \subset C,\) the normal cone of \(F\) is defined as the normal cone of any \(x\) belonging to the relative interior of \(F,\) viz \(N(C, F):= N(C, x).\) \textit{The normal cone lattice} \(\mathcal N(C)\) consists of all normal cones. For \(u \in \mathbb E,\) the touching cone of \(C\) along \(u,\; T(C, u)\) is the face of the normal cone \(N(C, F_{\perp}(C, u))\) which has \(u\) in the relative interior. The set of all touching cones \(\mathcal T (C)\) is called \textit{the touching cone lattice}. Each of these four lattices is ordered by inclusion and \(\mathcal F_{\perp}(C) \subset \mathcal F (C).\)NEWLINENEWLINEAmong the results, the author proves that the assignment of normal cones to exposed faces \(N(C): \mathcal F_{\perp}(C) \to \mathcal N (C), \; F \to N(C, F) \) is an antitone (order reversing) lattice isomorphism. Moreover, for a convex body \(K \subset \mathbb E\) with the origin in the interior, \(0 \in \text{int}\, (K)\), one defines the polar body \(K^{\circ} := \{ u \in \mathbb E: \langle u, x\rangle\;\leq 1, \; \forall x \in K \},\) whereas the positive hull \(pos (S)\) of a subset \(S \subset \mathbb E\) is defined as the set of all finite linear combinations with positive coefficients of elements of \(S\). Then the author establishes an isotone (order preserving) lattice isomorphism from the exposed faces of the polar body \(K^{\circ}\) to the normal cones of \(K,\) by \(\mathcal F_{\perp} (K^{\circ}) \to \mathcal N (K), \; F \to \text{pos}\, (F),\) which extends to an isomorphism \(\mathcal F (K^{\circ}) \to \mathcal T (K), \; F \to \text{pos}\, (F).\) The author also studies the compatibility of the four lattices under projections to or intersections with affine subspaces, proving that normal cones project to normal cones but exposed faces may project to non-exposed faces. Dually, exposed faces are preserved under intersections with affine subspaces whereas the property of being a normal cone is not. The author shows that the touching cones can detect exposed faces which are intersections of coatoms of the face lattice and they also relate to special smoothness in dimension two. In addition, the author considers an interesting example of a convex body (a state space) in the Euclidean space of Hermitian matrices for which \(\mathcal F (K) = \mathcal F_{\perp} (K)\) and \(\mathcal N (K) = \mathcal T (K).\)