Let P be a set of n points in Rd and F be a
family of geometric objects. We call a point xβP a strong centerpoint of
P w.r.t F if x is contained in all FβF 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. We
also prove the existence of strong centerpoints for abstract set systems with
bounded intersection