Let $s^{\prime}$ be another toric section of $L$. Then there is an element $m\in M$ such that $s^{\prime}=\chi^{m}s$. The corresponding virtual support function is $\Psi^{\prime}=\Psi-m$. Denote by $\|\cdot\|$ and $\|\cdot\|^{\prime}$ the metrics associated to $s,\Psi$ and to $s^{\prime},\Psi^{\prime}$ respectively. Then

$$\|s(p)\|^{\prime}=\|\chi^{-m}s^{\prime}(p)\|^{\prime}=\operatorname{e}^{\lambda_{K}(m+\Psi^{\prime})({\operatorname{val}}(p))}=\operatorname{e}^{\lambda_{K}\Psi({\operatorname{val}}(p))}=\|s(p)\|.$$

Thus both metrics agree. ∎