  Given any polar pair of convex bodies we study its conjugate face maps and we
characterize conjugate faces of non-exposed faces in terms of normal cones. The
analysis is carried out using the positive hull operator which defines lattice
isomorphisms linking three Galois connections. One of them assigns conjugate
faces between the convex bodies. The second and third Galois connection is
defined between the touching cones and the faces of each convex body
separately. While the former is well-known, we introduce the latter in this
article for any convex set in any finite dimension. We demonstrate our results
about conjugate faces with planar convex bodies and planar self-dual convex
bodies, for which we also include constructions.
