In the region $r(\bm{x})\leq 1/4$, we can relate the distance function $d_{\mathcal{P}}(\bm{x})\equiv d_{\omega_{T}}(\bm{x},\mathcal{P})$ on $\mathcal{M}_{T}$ with $r(\bm{x})=d_{Q}(\pi(\bm{x}),P)$ as follows,

(4.277)

$\displaystyle\begin{cases}C^{-1}\cdot T^{\frac{2-n}{2n}}\cdot r(\bm{x})^{1/2}\leq d_{P}(\bm{x})\leq C\cdot T^{\frac{2-n}{2n}}\cdot r(\bm{x})^{1/2},&r(\bm{x})\leq T^{-1},\\ C^{-1}\cdot T^{\frac{1}{n}}\cdot r(\bm{x})\leq d_{P}(\bm{x})\leq C\cdot T^{\frac{1}{n}}\cdot r(\bm{x}),&2T^{-1}\leq r(\bm{x})\leq\frac{1}{4}.\end{cases}$

The weight function we used in [HSVZ18] was defined with respect to the intrinsic distance function $d_{P}(\bm{x})$. Noticing by (4.277), the weight function defined by (4.273) essentially coincides with the one in [HSVZ18] (see Section 8 in [HSVZ18]).