Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.
WLOG let $A_1$ be the origin $0$.
Take any point $A_i$, then $P_i=A_i+P_1$, lies in $2 P_1$, the polyhedron $P_1$ stretched by the factor $2$ on $P_1=0$.
More general: take any $p,q$ in any convex shape $S$. Then $p+q \in 2S$.
Prove: since $S$ is convex, $\frac{p+q}{2} \in S$, thus $p+q \in 2S$.
Now all these nine polyhedrons lie inside $2 P_1$. Let $V$ be the volume of $P_1$.
Then some polyhedrons with total sum of volumes $9V$ lie in a shape of volume $8V$, thus they must overlap, meaning that they have an interior point in common.