The Delaunay oriented matroid is studied in detail by F. Santos (actually this paper is more general).
For a set of points $S$ (in any dimension), let $C$ be a sphere, $C^+$ be its interior, $C^-$ be its exterior, then $(S\cap C^+, S\cap C, S\cap C^-)$ is a covector of the Delaunay oriented matroid. The fact that it's an oriented matroid can be seen by lifting the points to a paraboloid in the space with one dimension higher and then observing a hyperplane arrangement.
Question: How did / will Delaunay oriented matroid help understanding Delaunay triangulations?