On Strong Centerpoints

Let P be a set of n points in R and F be a family of geometric objects. We call a point x ∈ P a strong centerpoint of P w.r.t F if x is contained in all F ∈ F that contains more than cn points from P , where c is a fixed constant. A strong centerpoint does not exist even when F is the family of halfspaces in the plane. We prove the existence of strong centerpoints with exact constants for convex polytopes defined by a fixed set of orientations and hyperplanes in R. We also prove the existence of strong centerpoints for abstract set systems with bounded intersection.