An inequality for polyhedra and ideal triangulations of cusped hyperbolic 3-manifolds

It is not known whether every noncompact hyperbolic 3-manifold of finite volume admits a decomposition into ideal tetrahedra. We give a partial solution to this problem: Let $M$ be a hyperbolic 3-manifold obtained by identifying the faces of $n$ convex ideal polyhedra $P_{1},\dots ,P_{n}$. If the faces of $P_{1},\dots ,P_{n-1}$ are glued to $P_{n}$, then $M$ can be decomposed into ideal tetrahedra by subdividing the $P_{i}$'s.