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. ∎