Let $diam(A)$ be the graph distance between the two farthest vertices contained in a finite set $A \subset \mathbb{Z}^d$. Is it true that there exists a real $\eta(d)>0$ such that for any finite $A \subset \mathbb{Z}^d$, it is possible to cover $A$ with a box of diameter $\eta \cdot diam(A)$?
If yes, what is the minimal $\eta$?
If not, which restrictions should we impose on the family of sets $A$ in order this to be true?