Volumes of balls in large Riemannian manifolds


We prove two lower bounds for the volumes of balls in a Riemannian manifold. If $(M^n, g)$ is a complete Riemannian manifold with filling radius at least $R$, then it contains a ball of radius $R$ and volume at least $\delta(n) R^n$. If $(M^n, \mathrm{hyp})$ is a closed hyperbolic manifold and if $g$ is another metric on $M$ with volume no greater than $\delta(n) \mathrm{Vol}(M, \mathrm{hyp})$, then the universal cover of $(M,g)$ contains a unit ball with volume greater than the volume of a unit ball in hyperbolic $n$-space.