We show that the planar dual to the Euclidean farthest point Voronoi diagram for the set of vertices of a convex polygon has the lexicographic minimum possible sequence of triangle angles, sorted from sharpest to least sharp. As a consequence, the sharpest angle determined by three vertices of a convex polygon can be found in linear time.

We introduce the concept of weighted skeleton of a polygon and present various decomposition and optimality results for this skeletal structure when the underlying polygon is convex.

The n-interior point variant of the Erdős-Szekeres problem is the following: for any n, n ≥ 1, does there exist a g(n) such that every point set in the plane with at least g(n) interior points has a convex polygon containing exactly n-interior points. The existence of g(n) has been proved only for n ≤ 3. In this paper, we show that, for point sets having at most logarithmic number of convex lay...

Given a convex polygon of n sides, one can draw n disks (called side disks) where each disk has a different side of the polygon as diameter and the midpoint of the side as its center. The intersection graph of such disks is the undirected graph with vertices the n disks and two disks are adjacent if and only if they have a point in common. We prove that for every convex polygon this graph is pl...

In this paper, we consider the continuous Fermat-Weber problem, where the customers are continuously (uniformly) distributed along the boundary of a convex polygon. We derive the closed-form expression for finding the average distance from a given point to the continuously distributed customers along the boundary. A Weiszfeld-type procedure is proposed for this model, which is shown to be linea...