First observe that the condition $l_{i}\in\mathbb{R}$ in the Archimedean case and $l_{i}\in\mathbb{Q}$ in the non-Archimedean case is equivalent to the condition $l_{i}\in\mathbb{Q}\,{\operatorname{val}}_{K}(K^{\times})$. Let $e>0$ be an integer such that $em_{i}\in M$ and $el_{i}\in{\operatorname{val}}_{K}(K^{\times})$ for $i=0,\dots,r$.

Consider the linear map $H\colon N_{\mathbb{R}}\to\mathbb{R}^{r}$ given by $H(u)=(em_{i}(u)-em_{0}(u))_{i=1,\dots,r}$ and the affine map $A=H+\boldsymbol{l}$ with $\boldsymbol{l}=(el_{i}-el_{0})_{i=1,\dots,r}$. By Lemma 3.79,

We claim that, for each $\sigma\in\Sigma$ there exists $\sigma_{i_{0}}\in\Sigma_{\Delta^{r}}$ such that $H(\sigma)\subset\sigma_{i_{0}}$. Indeed, $\Psi(u)=\min_{i}\{m_{i}(u)\}$. Since $\Psi$ is a support function on $\Sigma$, for each $\sigma\in\Sigma$, there exists an $i_{0}$ such that $\Psi(u)=m_{i_{0}}(u)$ for all $u\in\sigma$. Writing $e_{0}^{\vee}=0$, this condition implies

Hence, $H(\sigma)\subset\sigma_{i_{0}}$, where $\sigma_{i_{0}}\in\Sigma_{\Delta^{r}}$ is the cone $\{v|\min_{0\leq i\leq r}\{e_{i}^{\vee}(v)\}=e_{i_{0}}^{\vee}(v)\}$ and the claim is proved.

Therefore, we can apply Theorem 4.9 and given a point $p\in\mathbb{P}^{r}_{r}(K)$ such that ${\operatorname{val}}_{K}(p)=\boldsymbol{l}$, there is an equivariant map $\varphi_{p,H}\colon X_{\Sigma}\to\mathbb{P}^{r}$. By Example 4.44, there is an isomorphism $L^{\otimes e}\simeq\varphi_{p,H}^{*}\mathcal{O}(1)$ and $a\in K^{\times}$ with ${\operatorname{val}}_{K}(a)=l_{0}$ such that $(a^{-1}\chi^{-m_{0}}s)^{\otimes e}$ corresponds to $\varphi_{p,H}^{*}(s_{\Psi_{\Delta^{r}}})$.

Let ${\overline{L}}$ be the line bundle $L$ equipped with the metric induced by the above isomorphism and the canonical metric of $\mathcal{O}(1)^{{\text{\rm an}}}$. Then

as stated. ∎