Empty pseudo-triangles in point sets

We study empty pseudo-triangles in a set PP of nn points in the plane, where an empty pseudo-triangle has its vertices at the points of PP, and no points of PP lie inside. We give bounds on the minimum and maximum number of empty pseudo-triangles. If PP lies inside a triangle whose corners must be the convex vertices of the pseudo-triangle, then there can be between Θ(n2)Θ(n2) and Θ(n3)Θ(n3) empty pseudo-triangles. If the convex vertices of the pseudo-triangle are also chosen from PP, this number lies between Θ(n3)Θ(n3) and Θ(n6)Θ(n6). If we count only star-shaped pseudo-triangles, the bounds are Θ(n2)Θ(n2) and Θ(n5)Θ(n5). We also study optimization problems: minimizing or maximizing the perimeter or the area over all empty pseudo-triangles defined by PP. If PP lies inside a triangle whose corners must be used, we can solve these problems in O(n3)O(n3) time. In the general case, the running times are O(n6)O(n6) for the maximization problems and O(nlogn)O(nlogn) for the minimization problems.

► We study the minimum and maximum number of empty pseudo-triangles defined by a point set. ► We distinguish in star-shaped and general pseudo-triangles. ► We give algorithms to determine optimal pseudo-triangles defined by a point set.