Triangles in arrangements of lines. II

Abstract:

We show that given n lines in the real projective plane, no $n - 1$ of which are concurrent, the number ${p_3}$ of triangular regions formed is at most $\tfrac {2}{5}n(n - 1)$, equality being possible. We also show that if $n \geqslant 6$ then ${p_3} \leqslant \tfrac {7}{{18}}n(n - 1) + \tfrac {1}{3}$. Grünbaum has conjectured ${p_3} \leqslant \tfrac {1}{3}n(n - 1)$.