As $s\to+\infty$, the pushforward measure $(\text{Log}_{s})_{*}d\mu_{s}$ converges to the Lebesgue measure $d\mu_{\infty}$ supported on $\partial\Delta_{\lambda}^{\vee}$. In particular

$$\int_{X_{s}}d\mu_{s}\to\text{Vol}(\partial\Delta_{\lambda}^{\vee})=\int_{\partial\Delta_{\lambda}^{\vee}}d\mu_{\infty}.$$

(15)

Morever, there is a uniform exponential measure decay estimate

$$d\mu_{s}(\{z\in X_{s}:\text{dist}_{\mathbb{R}^{n+1}}(\text{Log}_{s}(z),\partial\Delta_{\lambda}^{\vee})>s^{-1}\Lambda\})\leq C^{\prime}e^{-C\Lambda},\quad\forall\Lambda>0.$$

(16)