On the polygonal diameter of the interior, resp. exterior, of a simple closed polygon in the plane

We give a tight upper bound on the polygonal diameter of the interior, resp. exterior, of a simple $n$-gon, $n\ge 3$, in the plane as a function of $n$, and describe an $n$-gon $(n\ge 3)$ for which both upper bounds (for the interior and the exterior) are attained \emph{simultaneously}.