The function $\psi_{\|\cdot\|}$ is given by

$$\psi_{\|\cdot\|}(u)=\begin{cases}m_{0}u-m_{0}a-\frac{\alpha_{0}}{e},&\text{ if }u\geq a,\\ \frac{(\alpha_{i+1}-\alpha_{i})u-(\alpha_{i+1}-\alpha_{i})(a-i)-\alpha_{i}}{e},&\text{ if }a-i\geq u\geq a-i-1,\\ m_{\infty}u-m_{\infty}(a-k)-\frac{\alpha_{k}}{e},&\text{ if }a-k\geq u.\end{cases}$$

In other words, if $\Pi$ is the polyhedral complex in $N_{\mathbb{R}}$ given by the intervals

$$(-\infty,a-k],\quad[a-i,a-i+1],\ i=1,\dots,k,\quad[a,\infty),$$

then $\psi_{\|\cdot\|}$ is the rational piecewise affine function on $\Pi$ characterized by the conditions

1. (1)

   $\operatorname{rec}(\psi_{\|\cdot\|})=\Psi$,

2. (2)

   the value of $\psi_{\|\cdot\|}$ at the point $a-i$ is $-\alpha_{i}/e$.