The nine Morse generic tetrahedra

From computational geometry comes the notion of a Gabriel graph of a point set in the plane. The Gabriel graph consists of those edges connecting two points of the point set such that the circle whose diameter is the edge does not contain any point of the point set in its interior. We define a generalization of the Gabriel graph to $n$ dimensions: the Morse poset. Using Morse theory we prove that for a generic set of 4 points in $\bR^3$ there are nine different Morse posets, up to combinatorial equivalence. At the end we mention some open questions and report on the results of computer experiments concerning these. We also compare our shape classification to another criterion widely used in computer science.