The Collapsing Walls Theorem

At first, this may seem obvious, but in fact the problem is already non-trivial even in the case of four-sided pyramids, which can possibly have some obtuse dihedral angles (see Figure 1). Formally, we have the following result: Collapsing Walls Theorem. Let P ⊂ R be a pyramid over a convex polygon Q. For a face F of P , denote by eF the edge between F and the base: eF = F ∩Q, and let AF denotes the result of rotation of F around eF in the direction of P , onto the plane which contains Q. Then Q ⊆ ∪F AF , where the union is over all faces F of P , different from Q.