  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.
