Curvature free volume estimates (Q801542)

Let M be a compact Riemannian manifold without boundary. Then the author proves among other things the following estimate of the average value of the volume of a ball B(x,R) of radius R around \(x: \alpha (n)R^ n/\pi^ n\leq Ave_{x\in M}[Vol(B(x,R))]\) where R is smaller than the injectivity radius of M and \(\alpha\) (n) is the volume of the unit n- sphere.