  We show that for a monic polynomial p of degree d, the length of the level
set {z: |p(z)|=1} is at most 9.2 d, which improves an earlier estimate due to
P. Borwein. For d=2 we show that the extremal level set is the Bernoullis'
Lemniscate. One ingredient of our proofs is the fact that for an extremal
polynomial this level set is connected.
