Optimal Empty Pseudo-Triangles in a Point Set

Given n points in the plane, we study three optimization problems of computing an empty pseudo-triangle: we consider minimizing the perimeter, maximizing the area, and minimizing the longest maximal concave chain. We consider two versions of the problem: First, we assume that the three convex vertices of the pseudotriangle are given. Let n denote the number of points that lie inside the convex hull of the three given vertices, we can compute the minimum perimeter or maximum area pseudo-triangle in O(n) time. We can compute the pseudo-triangle with minimum longest concave chain in O(n log n) time. If the convex vertices are not given, we achieve running times of O(n log n) for minimum perimeter, O(n) for maximum area, and O(n log n) for minimum longest concave chain. In any case, we use only linear space.