Empty Pseudo-Triangles in Point Sets

Abstract

We study empty pseudo-triangles in a set P of n points in the plane, where an empty pseudo-triangle has its vertices at the points of P, and no points of P lie inside. We give bounds on the minimum and maximum number of empty pseudo-triangles. If P lies inside a triangle whose corners must be the convex vertices of the pseudo-triangle, then there can be between Theta(n(2)) and Theta(n(3)) empty pseudo-triangles. If the convex vertices of the pseudo-triangle are also chosen from P, this number lies between Theta(n(3)) and Theta(n(6)). If we count only star-shaped pseudo-triangles, the bounds are Theta(n(2)) and Theta(n(5)). We also study optimization problems: minimizing or maximizing the perimeter or the area over all empty pseudo-triangles defined by P. If P lies inside a triangle whose corners must be used, we can solve these problems in O(n(3)) time. In the general case, the running times are O(n(6)) for the maximization problems and O(n log n) for the minimization problems.