Given a finite set X of points in Rn 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.
- Positive fraction intersection
- Selection theorem
- Weak epsilon-nets