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

TL;DR

This paper establishes the exact maximum polygonal diameter for the interior and exterior of any simple polygon with n sides in the plane, providing tight bounds and specific examples where these bounds are achieved.

Contribution

It presents the first tight upper bounds on the polygonal diameter of both the interior and exterior of simple polygons, along with examples attaining these bounds simultaneously.

Findings

Derived tight upper bounds for polygonal diameters.

Constructed polygons that achieve these bounds.

Provided a comprehensive analysis of polygonal diameters.

Abstract

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}.