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

We give a tight upper bound for the polygonal diameter (a.k.a. link diameter) of the interior, resp. exterior, of a simple nn-gon, n≥3n≥3, in the plane as a function of nn, and describe annn-gon (n≥3)(n≥3) for which both upper bounds (for the interior and for the exterior) are attained simultaneously.