ScalingStacks

Combining (3.5) with Cheeger-Colding’s segment inequality [4, Theorem 2.11] applied to the function $|d\rho_{t}|_{\omega_{t}}$ we obtain

(3.6)

$$\begin{split}D_{t}(\mu_{t}(A_{1})+\mu_{t}(A_{2}))\int_{X_{t}}|d\rho_{t}|_{\omega_{t}}d\mu_{t}&\geqslant C^{-1}\int_{A_{1}\times A_{2}}\left(\int_{\gamma_{x,y}}|d\rho_{t}|_{\omega_{t}}ds\right)d\mu_{x}d\mu_{y}\\ &\geqslant C^{-1}\int_{A_{1}\times A_{2}}(\rho_{t}(y)-\rho_{t}(x))d\mu_{x}d\mu_{y}\\ &\geqslant\frac{C^{-1}}{3}\mu_{t}(A_{1})\mu_{t}(A_{2}),\end{split}$$

where $D_{t}=\mathrm{diam}(X_{t},\omega_{t})$, and in the $\int_{A_{1}\times A_{2}}$ we are actually only integrating over the subset of pairs $(x,y)$ which are joined by a unique $\omega_{t}$-minimal geodesic, which has full measure (cf. [4]).