The Chebyshev center of a bounded set $Q$ having non-empty interior is defined in this question as the center of the minimal-radius ball enclosing the entire set $Q$.
Let $B$ be the minimum-volume axis-aligned box containing the set $Q$, $r_B$ the radius of the minimal-radius ball enclosing $B$ and $r_Q$ the radius of the minimal-radius ball enclosing $Q$.
My questions are the following:
- It is true that the Chebyshev center of $Q$ is always inside $B$?
- Given that (1) is true, is there any upper bound to $|r_B - r_Q|$?