Dushnik–Miller dimension of TD-Delaunay complexes

TD-Delaunay graphs, where TD stands for triangular distance, are a variation of the classical Delaunay triangulations obtained from a specific convex distance function (Chew and Drysdale, 1985). In Bonichon et al. (2010) the authors noticed that every triangulation is the TD-Delaunay graph of a set of points in R2, and conversely every TD-Delaunay graph is planar. It seems natural to study the generalization of this property in higher dimensions. Such a generalization is obtained by defining an analogue of the triangular distance for Rd. It is easy to see that TD-Delaunay complexes of Rd−1 are of Dushnik–Miller dimension d. The converse holds for d=2 or 3 and it was conjectured to hold for larger d (Mary, 2010). Here we disprove the conjecture already for d=4.