Corollary of Conjecture: .

Similarly the corollary, on small perturbations of *f* = 1, implies the sharp, known Wirtinger inequality:

for (nonperiodic) *u* of mean 0. (For *u* periodic the sharp constant is .)

The conjectured inequality says that for all ,

.

The case holds trivially; is the standard isoperimetric inequality. Both sides are concave as functions of . Would some kind of interpolation argument prove the result for all ?

]]>This does not of course help with the conjecture.

*Right you are, thanks very much. The undergraduates came upon it in a completely different way, as a generalization of the isoperimetric inequality (p=1) to sectors of planes with density , using the four vertex theorem. Your comment inspired a simpler geometric comparison argument. Do you believe the conjecture?—FM*