Dushnik-Miller dimension of TD-Delaunay complexes

Abstract

TD-Delaunay graphs, where TD stands for triangular distance, is a variation of the classical Delaunay triangulations obtained from a specific convex distance function. Bonichon et. al. 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 independently by Mary and Evans et. al. to hold for larger $d$. Here we disprove the conjecture already for $d=4$.

A short version appears in the proceedings of EuroCG 2017