Dense point sets have sparse Delaunay triangulations

Abstract 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 p,.s with spread A has complexity,O(AS}. This bound,is tight in the worst case for all A = O(v~). In particular, the Delaunay triangulation of any dense point set has linear complexity. On the other hand, for any n and A = O{n), we construct a regular triangulation of complexity,~{nA} whose n. vertices have spread A.