What does v mean in the following, does it a point inside the polyhedron?
There may be different interpretations, but the shading of the images suggests that the polyhedron $P$ is considered as the set of all points inside linear faces, including the faces themselves. Thus the shaded areas (including their boundaries) depict the polyhedra $CH(U)$ etc. If the full 10gon were shaded, that would depict the full polyhedron $P$.
By the way, it seems that the statements in the text may not be correct unless one restricts the discourse to convex polyhedra, that is the intersection of finitely many half-spaces. Then of course $v\in P$ simply means that $v$ is in each of the half-spaces, that is all defining inequalities of P$ are fulfilled.