Dense point sets have sparse delaunay triangulations or "... but not too nasty"

The spread of a finite set of points is the ratio between the longest and shortest pairwise distances. We prove that the Delaunay triangulation of any set of n points in ℝ³ with spread Δ has complexity O(Δ³). This bound is tight in the worst case for all Δ = O(√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 Δ = O(n), we construct a regular triangulation of complexity Ω(nΔ) whose n vertices have spread Δ.