Remarks on the van der Waerden permanent conjecture

The van der Waerden permanent conjecture is shown to belong to a large family of conjectured inequalities which are of some interest in themselves and all of which might be provable by a routine computation with convex bodies. In fact, the permanent conjecture in cases n=3 and 4 does yield to this method. For n=5, with the computations made by hand, no proof was found, but a slight extension of the computation (which would probably require electronic assistance) could still settle this case and perhaps even a few more small values of n. The question of whether the method must work remains open.