Minimal simplicial dissections and triangulations of convex 3-polytopes

This paper addresses three questions related to minimal triangulations of a three-dimensional convex polytope P. Can the minimal number of tetrahedra in a triangulation be decreased if one allows the use of interior points of P as vertices? Can a dissection of P use fewer tetrahedra than a triangulation? Does the size of a minimal triangulation depend on the geometric realizatio n of P? The main result of this paper is that all these questions have an affirmati ve answer. Even stronger, the gaps of size produced by allowing interior ve rtices or by using dissections may be linear in the number of points.