<italic>We show that for any set &Pgr; of n points in three-dimensional space there is a set Q of &Ogr;(n<supscrpt>1/2</supscrpt> log<supscrpt>3</supscrpt> n) points so that the Delaunay triangulation of &Pgr; ∪ Q has at most &Ogr;(n<supscrpt>3/2</supscrpt> log<supscrpt>3</supscrpt> n) edges — even though the Delaunay triangulation of &Pgr; may have &OHgr;(n<supscrpt>2</supscrpt>) 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</italic> &OHgr;(<italic>m</italic><supscrpt>2</supscrpt>/<italic>n</italic><supscrpt>2</supscrpt> log<supscrpt>3</supscrpt> <italic>n</italic><supscrpt>2</supscrpt>/<italic>m</italic>) <italic>of the spheres in S.</italic>
par><italic>Our results generalize to arbitrary fixed dimensions, to geometric bodies other than spheres, and to geometric structures other than Delaunay triangulations.</italic>