Computing the link center of a simple polygon

The link center of a simple polygon P is the set of points x inside P at which the maximal link-distance from x to any other point in P is minimized, where the link distance between two points x, y inside P is defined as the smallest number of straight edges in a polygonal path inside P connecting x to y. We prove several geometric properties of the link center and present an algorithm that calculates this set in time <italic>&Ogr;</italic> (<italic>n</italic><supscrpt>2</supscrpt>), where <italic>n</italic> is the number of sides of P. We also give an <italic>&Ogr;</italic>(<italic>n</italic> log <italic>n</italic>) algorithm for finding a point x in an approximate link center, namely the maximal link distance from x to any point in P is at most one more than the value attained from the link center.