I have the following problem. Suppose to have a cube $Q$ and a symmetric box type ball $B$ whose center is outside Q, but such that $Q\cap B \ne\emptyset$. Furthermore $B$ is not oriented like $Q$. I want to prove that we can translate the ball $B$ - we denote the translated version with $B'$ - in such a way that $B'$ still covers the intersection $B\cap Q$ but this time the center of $B'$ is contained in $B\cap Q$. You can find a figure describing the situation here. Do you have any idea on how I can prove this result? I tried the following approach but now I am stuck. Since the $Q\cap B \ne\emptyset$ there exists a point $p\in Q\cap B$. If this are no other points in $Q\cap B$, we have finished, otherwise we should translate $B'$ along the direction vector connecting the center of $B$ with a sort of "center" of $Q\cap B$. Can you help me to understand how this point looks like?
Thanks in advance, Best,
Giorgio