Cell decomposition of polytopes by bending

If a lineL crosses a polygonP, then bendingP up on both sides ofL yields a 3-polytope whose “upper” boundary projects back to a cell decomposition ofP transverse toL, and to a triangulation in the general case. We generalize this simple idea to polytopes of arbitrary dimension, and use it to answer several questions posed recently about possible decompositions of polytopes and of regions between polytopes.