To appear in Discrete & Computational Geometry.
Abstract:
The spread of a set of points is the ratio between the largest ans smallest pairwise distances. We show that the Delaunay triangulation of any set of n points in R^{3} with spread D has complexity O(D^{3}). This bound is tight in the worst case for all D=O(sqrt{n}). In particular, the Delaunay triangulation of any dense point set has linear complexity. We also generalize this upper bound to regular triangulations of k-ply systems of balls, unions of several dense point sets, and uniform samples of smooth surfaces. On the other hand, for any n and any D=O(n), we construct a regular triangulation of complexity Omega(nD) whose n vertices have spread D.