Considering the hypothesis that there exists a polyhedron with a minimal triangulation by $2n-10$ tetrahedra, earlier results show that such polyhedra can have only vertices of order 5, 6 or separated vertices of order 4. Other polyhedra have minimal triangulation with a smaller number of tetrahedra. This paper presents the examples of polyhedra with the mentioned property and with the triangulation by $2n-11$ tetrahedra.