  Evolving smooth, compact hypersurfaces in R^{n+1} with normal speed equal to
a positive power k of the mean curvature improves a certain 'isoperimetric
difference' for k >= n-1. As singularities may develop before the volume goes
to zero, we develop a weak level-set formulation for such flows and show that
the above monotonicity is still valid. This proves the isoperimetric inequality
for n <= 7. Extending this to complete, simply connected 3-dimensional
manifolds with nonpositive sectional curvature, we give a new proof for the
Euclidean isoperimetric inequality on such manifolds.
