I am currently working on a problem involving algebraic geometry and as a part of the research it would be helpful for me to understand the process of alternation, also called partial truncation, particularly as it relates to the hypercube, but also for other polytopes.
Coxeter states in Regular Polytopes section 8-6 that the rejected corners in the alternation are cut off by hyperplanes parallel to those of the corresponding vertex figures, and that these slices introduce no new vertices. This is also referenced in the paper Homology Representations Arising from the Half Cube by R.M. Green in section 2.3, who states that the process of alternation does not allow for the introduction of new vertices in the creation of the demihypercube.
It is clear to me that in the case of the cube no new vertices are introduced, but how can that idea be expanded to hypercubes of higher degree and other polytopes? How can we be sure that no new vertices are created by the new hyperplanes? It seems as if there must be a simple explanation as both Coxeter and Green take the statement as fact, and I would appreciate help in finding the answer.