A Note on Touching Cones and Faces

We study touching cones of a (not necessarily closed) convex set in a finite-dimensional real Euclidean vector space and we draw relationships to other concepts in Convex Geometry. Exposed faces correspond to normal cones by an antitone lattice isomorphism. Poonems generalize the former to faces and the latter to touching cones, these extensions are non-isomorphic, though. We study the behavior of these lattices under projections to affine subspaces and intersections with affine subspaces. We prove a theorem that characterizes exposed faces by assumptions about touching cones. For a convex body K the notion of conjugate face adds an isotone lattice isomorphism from the exposed faces of the polar body Ko to the normal cones of K. This extends to an isomorphism between faces and touching cones.