  Let $P$ be a set of $n$ points in $\mathbb{R}^d$ and $\mathcal{F}$ be a
family of geometric objects. We call a point $x \in P$ a strong centerpoint of
$P$ w.r.t $\mathcal{F}$ if $x$ is contained in all $F \in \mathcal{F}$ that
contains more than $cn$ points from $P$, where $c$ is a fixed constant. A
strong centerpoint does not exist even when $\mathcal{F}$ 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.
