  We show how 'test' vector fields may be used to give lower bounds for the
Cheeger constant of a Euclidean domain (or Riemannian manifold with boundary),
and hence for the lowest eigenvalue of the Dirichlet Laplacian on the domain.
Also, we show that a continuous version of the classical Max Flow Min Cut
Theorem for networks implies that Cheeger's constant may be obtained precisely
from such vector fields. Finally, we apply these ideas to reprove a known lower
bound for Cheeger's constant in terms of the inradius of a plane domain.
