Abstract
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 $\mathbb{R}^2$, 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 $\mathbb{R}^d$. It is easy to see that TD-Delaunay complexes of $\mathbb{R}^{d-1}$ 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 $d = 4$.