slimming down by adding; selecting heavily covered points
We show that for any set &Pgr; of n points in three-dimensional space there is a set Q of &Ogr;(n 1/2 log 3 n) points so that the Delaunay triangulation of &Pgr; ? Q has at most &Ogr;(n 3/2 log 3 n) edges  even though the Delaunay triangulation of &Pgr; may have &OHgr;(n 2 ) edges. The main tool of our construction is the following geometric covering result: For any set &Pgr; of n points in three-dimensional space and any set S of m spheres, where each sphere passes through a distinct point pair in &Pgr;, there exists a point x, not necessarily in &Pgr;, that is enclosed by &OHgr;( m 2 / n 2 log 3 n 2 / m ) of the spheres in S. par> Our results generalize to arbitrary fixed dimensions, to geometric bodies other than spheres, and to geometric structures other than Delaunay triangulations.
select point space triangulation tool general