In Riemannian geometry, the Bishop-Gromov theorem states that for an $n$-dimensional manifold $M$ with nonnegative Ricci curvature $Rc\geq 0$, then the volume quotient $$ \frac{\text{Vol} B(x,r)}{\omega_n r^n} $$ is nonincreasing for $r\in (0,\infty)$. In particular, the above gives $$ \frac{\text{Vol} B(x,r)}{\omega_n r^n}\leq 1 \;\text{ for all } r>0. $$
There have been several papers (e.g., Petersen/Wei "Relative Volume Comparison with Integral Curvature Bounds 1997) which generalize this to manifolds which satisfy the negative part of the Ricci tensor being $L^p$ small. For example, the aforementioned paper's main result gives the inequality $$ \frac{\text{Vol} B(x,r)}{\omega_n r^n} \leq (1+C \|Rc_-\|_p ^2)^{2p} $$ where $p\in (n/2,\infty)$ and $Rc_-=\max\{0,-\lambda_1\}$ where $\lambda_1$ is the smallest eigenvalue of $Rc$.
Thus we have an upper bound of the volume of a ball in terms of the integral of the curvature. But is there a similar result for a lower bound of the volume as well?
