# 4-polytopes with only one kind of regular facet

Is there a neat way to show (or a reference that already proves) that

• the 4-cube is the only convex 4-polytope in which all facets are regular 3-cubes?
• the 24-cell is the only convex 4-polytope in which all facets are regular octahedra?
• the 120-cell is the only convex 4-polytope in which all facets are regular dodecahedra?

Note that I do require that the facets are regular, so e.g. just any cubical polytope does not fit.

I am aware of Gosset's semiregular polytopes (vertex-transitive and all facets are regular polytopes), for which this statement is true, but I do not require vertex-transitivitiy here.

However, if my statement turns out to be wrong in this general form, then I wonder whether it holds when I require that all vertices are on a common sphere.

This is Theorem 1 (actually, Satz 1) of Roswitha Blind, Konvexe Polytope mit kongruenten regulären $$(n- 1)$$-Seiten im $$\Bbb{R}^n$$ $$(n \ge 4)$$, Comment. Math. Helvetici 54 (1979) 304--308. The short proof follows from two short lemmas, one of which cites Coxeter's Regular Polytopes and an article by Shephard.