I am facing a problem. Suppose to have a cube $Q=[0,1]^d$ and another cube $Q^{'}$ which is a circumscribed cube with different orientation. Let us consider a $\epsilon$ covering of $Q^{'}$ with kind of $||\cdot||\infty$ boxes oriented according to $Q^{'}$. Can we translate these covering boxes so that their centers are contained in $Q$ and they are a covering of $Q$? My guess is that the answer is yes thanks to convexity and symmetry of the cubes and the boxes but I am not sure how to prove it. I tried finding a common rule to translate boxes centered outside $Q$, but I didn't find a common rule to do that. Do you have any suggestion on how to prove this result?
Thanks!