On the Size of Higher-Dimensional Triangulations

Summary

I show that there are sets of n points in three dimensions, in general position, such that any triangulation of these points has only O(n^5/3) simplices. This is the first nontrivial upper bound on the MinMax triangulation problem posed by Edelsbrunner, Preparata and West in 1990: What is the minimum over all general-position point sets of the maximum size of any triangulation of that set? Similar bounds in higher dimensions are also given.

1. Introduction

In the plane, all triangulations of a set of points use the same number of triangles. This is a simple consequence of each triangle having an interior angle sum of π, and each interior point of the convex hull contributing an angle sum of 2π, which must be used up by the triangles.

Neither the constant size of triangulations nor the constant angle sum of simplices holds in higher dimensions. A classic example is the cube, which can be decomposed in two ways: into five simplices (cutting off alternate vertices) or into six simplices (which are even congruent; it is a well-known simple geometric puzzle to assemble six congruent simplices, copies of conv((000), (100), (010), (011)), into a cube).

For higher-dimensional cubes, the same problem was studied in a number of papers [Böhm 1989; Broadie and Cottle 1984; Haiman 1991; Hughes 1993; Hughes 1994; Lee 1985; Marshall 1998; Orden and Santos 2003; Sallee 1984; Smith 2000]. This suggest that one should be interested in the possible values of the numbers of simplices for arbitrary point sets.