One cannot always triangulate a non-convex polyhedron using only its vertices, sometimes one need to add more of them. A simple example of this phenomenon isĀ Schonhardt polyhedron. Here is a picture illustrating how one builds it that I drew for a forthcoming paper, using Tikz LaTeX package, which is awesome, but totally overwhelming.

It fact, it’s easy to see that the 6 vertices and 12 edges it has are not enough. Indeed, each pair of non-intersecting edges determines a simplex, but it’s easy to observe that any such selection will include one the forbidden pairs of vertices AC, A’B, or B’C’. (The LaTeX source of the picture is here).

The paper I mention is related to a topic of the program on inverse moment problems at IMS (NUS/Singapore) in late 2013-early 2014 which I co-organize.