Volume growth of some uniformly contractible Riemannian manifolds

It is shown that if a uniformly contractible Riemannian n-manifold (M,g) is K-quasi-isometric to an n-dimensional normed space((V^{n},|cdot|)), (K ≥  1), then(liminf_{Rrightarrow infty}frac{{Vol}_g( {Ball}_{R})}{R^{n}omega_{n}}geqfrac{1}{K^{2n}}) where ω _n is the volume of the unit Euclidean ball. In particular, if M is uniformly contractible and(d_{GH}((M,d_g), (V^n,|cdot|)) < infty ), then M has at least Euclidean volume growth. This corollary covers an earlier result by Burago and Ivanov. Our results are motivated by a volume growth theorem contained in Gromov’s book [Gromov in Progress in Mathematics, vol. 152, Birkhäuser, Boston, 1999, p. 256], we give a detailed proof of this theorem. Using the same argument, we also derive a generalization of the theorem which is pointed out by Gromov.