  If (M^n, g) is a complete Riemannian manifold with filling radius at least R,
then we prove that it contains a ball of radius R and volume at least c(n)R^n.
If (M^n, hyp) is a closed hyperbolic manifold and if g is another metric on M
with volume at most c(n)Volume(M,hyp), then we prove that the universal cover
of (M,g) contains a unit ball with volume greater than the volume of a unit
ball in hyperbolic n-space.
