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If the Ricci curvature of a compact Riemannian manifold of demsnion $n$ is greater than 1-n, does it follow that the volume entropy satisfies $$\liminf_{r\rightarrow \infty} \frac{\log vol B_r(p)}{r}\leq n-1$$

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Yes, this follows from the Bishop-Gromov volume comparison theorem.

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