Assume that $\|\cdot\|$ is semipositive. Let $u_{0}$ be a point of $N_{\mathbb{Q}}$ and let $v_{0}\in N$ be primitive. Since the condition of being concave is closed, if we prove that, for all choices of $u_{0}\in N_{\mathbb{Q}}$ and $v_{0}\in N$, the restriction of $\psi_{\|\cdot\|}$ to the line $u_{0}+\mathbb{R}v_{0}$ is concave, we will deduce that the function $\psi_{\|\cdot\|}$ is concave. Let $e\in\mathbb{N}^{\times}$ such that $eu_{0}\in N$. Then $H=K(\varpi^{1/e})$ is a finite extension of $K$ and there is a unique extension of the absolute value of $K$ to $H$. We will denote with ^′ the objects obtained by base change to $H$. Let $p\in X_{0,H}(H)$ such that ${\operatorname{val}}_{H}(p)=eu_{0}$. We consider the affine map $A\colon\mathbb{Z}\to N$ given by $l\mapsto v_{0}l+eu_{0}$, and let $H$ be the linear part of $A$. We consider the equivariant morphism $\varphi=\varphi_{p,H}\colon\mathbb{P}^{1}_{H}\to X_{\Sigma,H}$ of Theorem 4.9. The metric $\|\cdot\|$ induces an algebraic semipositive metric $\varphi^{\ast}\|\cdot\|^{\prime}$ on the restriction of $L^{\prime}$ (the line bundle obtained from $L$ by base change to $H$) to $\mathbb{P}^{1}_{H}$. By propositions 5.24 and 5.53(3) we obtain that

$$\psi_{\varphi^{\ast}\|\cdot\|}(u)=e\psi_{\|\cdot\|}(u_{0}+e^{-1}uv_{0}).$$

By Corollary 5.66 the left-hand side function is concave. Thus the restriction of $\psi_{\|\cdot\|}$ to $u_{0}+\mathbb{R}v_{0}$ is concave. We conclude that $\psi=\psi_{\|\cdot\|}$ is concave. ∎

By Proposition 5.67, the function $\psi_{\|\cdot\|}$ is concave. By Proposition 5.52 it is also rational piecewise affine. By Corollary 5.47, the metric $\|\cdot\|_{\mathbb{S}}=\|\cdot\|_{\psi}$ is toric algebraic and semipositive. ∎

Since the metric is semipositive and toric, by Proposition 5.67 the function $\psi$ is concave. Since, moreover it is algebraic, by Theorem 5.49 it is defined by a toric model $({\mathcal{X}}_{\Pi},D_{\psi},e)$ of $(X_{\Sigma},D_{\Psi})$ in the equivalence class determined by $\psi$. As in Remark 4.66, the irreducible components of ${\mathcal{X}}_{\Pi,o}$ are in bijection with the vertices of $\Pi$. For each vertex $v\in\Pi^{0}$, let $\xi_{v}$ be the point of $X_{\Sigma}^{{\text{\rm an}}}$ corresponding to the generic point of $V(v)$ defined by equation (2.15). Then, by equation (2.29),

$$c_{1}(\overline{L})^{n}\land\delta_{X_{\Sigma}}=\frac{1}{e^{n}}\sum_{v\in\Pi^{0}}\nu_{v}\deg_{D_{\psi}}V(v)\delta_{\xi_{v}}.$$

Thus, by Corollary 5.40,

$$({\operatorname{val}}_{K})_{\ast}(c_{1}(\overline{L})^{n}\land\delta_{X_{\Sigma}})=\frac{1}{e^{n}}\sum_{v\in\Pi^{0}}\nu_{v}\deg_{D_{\psi}}V(v)\delta_{v}.$$

But, using Proposition 3.95 and Proposition 4.105, the Monge-Ampère measure is given by

	$\displaystyle\mathcal{M}_{M}(\psi)$	$\displaystyle=\frac{1}{e^{n}}\mathcal{M}_{M}(e\psi)$
		$\displaystyle=\frac{1}{e^{n}}\sum_{v\in\Pi^{0}}\operatorname{vol}_{M}(v^{\ast})\delta_{v}$
		$\displaystyle=\frac{1}{n!e^{n}}\sum_{v\in\Pi^{0}}\nu_{v}\deg_{D_{\psi}}V(v)\delta_{v}.$

Since $\mathcal{M}_{M}(\psi)$ is a finite sum of Dirac deltas, we obtain that

$${\overline{\mathcal{M}}}_{M}(\psi)=\frac{1}{n!e^{n}}\sum_{v\in\Pi^{0}}\nu_{v}\deg_{D_{\psi}}V(v)\delta_{v}.$$

Hence we have proved (5.71). To prove equation (5.72) we just observe that $x_{v}=(\theta_{0}\circ{\operatorname{\mathbf{e}}}_{K})(v)$. ∎