Selecting Heavily Covered Points

A collection of geometric selection lemmas is proved, such as the following: For any set $P$ of $n$ points in three-dimensional space and any set ${\cal S}$ of $m$ spheres, where each sphere passes through a distinct point pair in $P$, there exists a point $x$, not necessarily in $P$, that is enclosed by $\Omega (m^2/(n^2 \log^6 {n^2 \over m}))$ of the spheres in ${\cal S}$. Similar results apply in arbitrary fixed dimensions, and for geometric bodies other than spheres. The results have applications in reducing the size of geometric structures, such as three-dimensional Delaunay triangulations and Gabriel graphs, by adding extra points to their defining sets.