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 Θ ( n 2 ) and Θ ( n 3 ) empty pseudo-triangles. If the convex vertices of the pseudo-triangle are also chosen from P , this number lies between Θ ( n 3 ) and Θ ( n 6 ) . If we count only star-shaped pseudo-triangles, the bounds are Θ ( n 2 ) and Θ ( 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. ► 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.