Since $\|\cdot\|$ is semipositive, $c_{1}(L,\|\cdot\|)\land\delta_{X_{\Sigma}}$ is a positive measure. By Corollary 5.65, $c_{1}(L,\|\cdot\|_{\mathbb{S}})\land\delta_{X_{\Sigma}}$ is a positive measure. Hence $\|\cdot\|_{\mathbb{S}}$ is a semipositive toric metric. By equation (5.64) and Lemma 5.60, the positivity of $c_{1}(L,\|\cdot\|_{\mathbb{S}})\land\delta_{X_{\Sigma}}$ implies that the function $\psi_{\|\cdot\|}=\psi_{\|\cdot\|_{\mathbb{S}}}$ is concave. ∎