Question: Show that polytopes which are both simple and simplicial are either simplicies or two-dimensional $n$-gons.
- A polytope $P$ is simple if every vertex is contained in exactly $d$ facets.
- A polytope $P$ is simplicial if every face is a simplex, that is, each facet of $P$ contains exactly $d$ vertices.
Attempt: The case for $d=2$ is immediate, so assume that $d\geq 3$. It suffices to show that if $P$ is not a simplex, then $P$ is an $n$-gon (i.e. $d$=2). Since $P$ is not a simplex, it contains at least $d+2$ vertices. At this point I want to use some sort of Pigeonhole principle argument to assert that at least one of the facets of $P$ contains at least $d+1$ vertices, contradicting that $P$ is simplicial. However, I am stuck on the precise argument to be used.
Would appreciate any help!