Back Faces of a Face Polytope

Full-dimensional cones of polytopal fans are spanned by invisible facets of face polytopes. However invisible faces cannot perform the same function for cones of lower dimension. It makes difficult considering fans with low-dimensional cones in any consistent manner. The paper defines back faces of a polytope so that they can be used as faces spanning cones of a polytopal fan regardless of dimension. As an application of back faces a theorem is proved according to which some positive combination of solid face polytopes flattens, becomes a plane face polytope.