  Given a finite set $X$ of points in $R^n$ and a family $F$ of sets generated
by the pairs of points of $X$, we determine volumetric and structural
conditions for the sets that allow us to guarantee the existence of a
positive-fraction subfamily $F'$ of $F$ for which the sets have non-empty
intersection. This allows us to show the existence of weak epsilon-nets for
these families. We also prove a topological variation of the existence of weak
epsilon-nets for convex sets.
