On Delaunay oriented matroids for convex distance functions

Abstract

For any finite point set S in Ed, an oriented matroid DOM(S) can be defined in terms of how S is partitioned by Euclidean hyperspheres. This oriented matroid is related to the Delaunay triangulation of S and is realizable, because of the lifting property of Delaunay triangulations. We prove that the same construction of a Delaunay oriented matroid can be performed with respect to any smooth, strictly convex distance function in the plane E2 (Theorem 3.5). For these distances, the existence of a Delaunay oriented matroid cannot follow from a lifting property, because Delaunay triangulations might be nonregular (Theorem 4.2(i). This is related to the fact that the Delaunay oriented matroid can be nonrealizable (Theorem 4.2(ii)).