# A directionally restricted version of Bishop-Gromov inequality

Recently, I'm reading the paper "On the structure of spaces of Ricci curvature bounded below I" by J.Cheeger& T.Colding, I'm confused with the formula (2.20):

Suppose $$(M,p)$$ is a complete n-dim Riemannian manifold with $$\operatorname{Ric}\geq -(n-1)$$. Put $$A_{r_1,r_2}(p)=B_{r_2}(p)\backslash \overline{B_{r_1}(p)}$$. Fix $$0 and $$0<\eta. Let $$X(p,r_1,,r_2,\eta)=\{q\in A_{r_1,r_2}(p)\mid \text{there exists a geodesic } \gamma \text{ with } \gamma(0)=p,\gamma(t)=q \text{ such that } \gamma|_{[0,t]} \text{ is minimal and } \gamma|_{[0,t+\eta]} \text{ is not minimal}\}$$

The authors claimed that "It follows directly from the directionally restricted version of Bishop-Gromov inequality that:" $$\frac{\operatorname{Vol}(X(p,r_1,r_2,\eta))}{\operatorname{Vol}(B_p(1))}\leq C(n,r_1,r_2)\eta.$$ where $$C=C(n,r_1,r_2)$$ is a constant depending only on $$n,r_1$$ and $$r_2$$. I wonder why this holds and how directionally restricted version of Bishop-Gromov inequality is applied.