A finite collection of simple
closed curves in the real projective plane each two of which have exactly one point in
common at which point they “cross” is called a pseudoline arrangement. Such
arrangements seem to have been first systematically studied by Levi. Recently B.
Grünbaum has called attention to the desirability for a better understanding of the
differences as well as the similarities in the behavior of arrangements of
lines and the arrangements of pseudolines. Among other things, Grünbaum
asks if every pseudoline arrangement, not all curves of which intersect in a
single point, must have a simple vertex (a point on exactly two of the curves
of the arrangement) as is the case in line arrangements. In fact Kelly and
Moser have shown that, in general, an n-line arrangement in an ordered
projective plane has at least 3∕7n simple vertices. It is shown here that their
reasoning can be nearly dualized to prove the analogous result for pseudoline
arrangements.