Computing external farthest neighbors for a simple polygon

Let 𝒫 be (the boundary of) a simple polygon with n vertices. For a vertex p of 𝒫, let ø(p) be the set of points on 𝒫 that are farthest from p, where the distance between two points is the length of the (Euclidean) shortest path that connects them without intersecting the interior of 𝒫. In this paper, the authors present an O(nlogn) algorithm to compute a member of ø(p) for every vertex p of 𝒫. As a corollary, the external diameter of 𝒫 can also be computed in the same time.