  We critically analyze the problem of formulating duality between fringe
visibility and which-way information, in multibeam interference experiments. We
show that the traditional notion of visibility is incompatible with any
intuitive idea of complementarity, but for the two-beam case. We derive a
number of new inequalities, not present in the two-beam case, one of them
coinciding with a recently proposed multibeam generalization of the inequality
found by Greenberger and YaSin. We show, by an explicit procedure of
optimization in a three-beam case, that suggested generalizations of Englert's
inequality, do not convey, differently from the two-beam case, the idea of
complementarity, according to which an increase of visibility is at the cost of
a loss in path information, and viceversa.
