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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 theinterior and the exterior) are attained \emph{simultaneously}.

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