Context for question:
The study of polyhedra and more generally of polytopes has never been particularly focused on rigor, and many references for results are often either non-existent, or impossible to find.
This lead me to try to prove some basic results on polyhedra. To be clear, the definition of polyhedra I use is a set of polygonal faces and edges such that two faces are adjacent to each edge, no two elements coincide and such that no subset of faces and edges forms a valid polyhedron (this excludes compounds).
The problem with answering the question:
With this in mind, I can now use the common definition of a regular polyhedron as a vertex-transitive polyhedron with congruent, regular faces. However, I ran into a problem when trying to prove that there were only 5 convex and 4 non-convex types (Platonic and Kepler-Poinsot solids):
First of all, the common proof for the Platonic solids (the one that uses the fact that you can't have many shapes with many sides around a vertex) assumes too many things. For example, one first needs to prove that the sum of the angles around a vertex is less than $2\pi$, which is false in the general, not-necessarily-convex case. Also, one needs to prove that given a vertex arrangement, there's at most one regular polyhedron that can be made. And even with all of this, the argument can't be generalized to non-convex solids: One can fit perfectly 7 equilateral triangles around a vertex, if you allow them to intersect.
So, why are there only 9 regular types of polyhedra?
Using Arentino's answer, I can immediately characterize the convex cases. If I could prove that the convex hull of any regular solid is regular, I could easily just check for polygons on the vertices of these five solids and check how to connect them to create the other four cases. However, I don't know why should this be the case either, and I have no idea of how to prove it.
Assuming all "known" properties of convex hulls (they are polyhedra for finite sets of points, etc.), I have been able to prove that the convex hulls have to be vertex-transitive (any symmetry of the original polyhedron that takes vertex A to B, will preserve the positions of the vertices as a whole and therefore the convex hull. As a consequence, the vertex figures are all congruent. However, I still need to prove that the faces are all congruent and regular, and I don't know how to do this. (Yet again).
At last, some useful literature! I found the following book (p. 260) where it describes why every Kepler-Poinsot solid must be a stellation of a regular polyhedron. (Although if someone can prove the thing about convex hills, I'd appreciate it deeply). There's just a tiny problem. I don't understand the proof completely. For example, why should the "kernel" of the polyhedron be a convex polyhedron? And even if it is, how did he show it consisted of regular faces?