On indecomposable polyhedra and the number of interior Steiner points

Abstract

The existence of 3d indecomposable polyhedra, that is, the interior of every such polyhedron cannot be decomposed into a set of tetrahedra whose vertices are all of the given polyhedron, is well-known. While the geometry and combinatorial structure of such polyhedra are much less studied. In this article, we first investigate the geometry of some wellknown examples, the so-called Schönhardt polyhedron [Schönhardt, 1928] and the Bagemihl's generalization of it [Bagemihl, 1948], which will be called Bagemihl polyhedra. We provide a construction of an interior point, so-called Steiner point, which can be used to tetrahedralize the Schönhardt and the Bagemihl polyhedra. We then provide a construction of a larger class of three-dimensional indecomposable polyhedra which often appear in grid generation problems. We show that such polyhedra have the same combinatorial structure as the Schönhardt and Bagemihl polyhedra, but they may need more than one interior Steiner point to be tetrahedralized. Given such a polyhedron with n ≥ 6 vertices, we show that it can be tetrahedralized by adding at most ... interior Steiner points. We also show that this number is optimal in the worst case.