Empty pseudo-triangles in point sets

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 @Q(n^2) and @Q(n^3) empty pseudo-triangles. If the convex vertices of the pseudo-triangle are also chosen from P, this number lies between @Q(n^3) and @Q(n^6). If we count only star-shaped pseudo-triangles, the bounds are @Q(n^2) and @Q(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(nlogn) for the minimization problems.

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