An Optimal Generalization of the Centerpoint Theorem, and its Extensions

We prove an optimal generalization of the centerpoint theorem: given a set P of n points in the plane, there exist two points (not necessarily among input points) that hit all convex objects containing more than 4n/7 points of P. We further prove that this bound is tight. We get this bound as part of a more general procedure for finding small number of points hitting convex sets over P , yielding several improvements over previous results