Let $f\in\mathscr{P}(N_{\mathbb{R}})$ and let $f(u)=\min_{0\leq i\leq r}(a_{i}(u)+\alpha_{i})$ be an H-representation of $f$. Write $\boldsymbol{\alpha}=(\alpha_{i}-\alpha_{0})_{i=1,\dots,r}$, and consider the linear map $H\colon N_{\mathbb{R}}\to\mathbb{R}^{r}$ given by $H(u)=(a_{i}(u)-a_{0}(u))_{i=1,\dots,r}$ and the affine map $A=H+\boldsymbol{\alpha}.$ Then

1. (1)

   $f=A^{\ast}\Psi_{\Delta^{r}}+a_{0}+\alpha_{0};$

2. (2)

   $f^{\vee}=\tau_{a_{0}}(H^{\vee})_{\ast}(\iota_{\Delta^{r}}-\boldsymbol{\alpha})-\alpha_{0}.$

This second function can be alternatively described as the function which parameterizes the upper envelope of the extended polytope

$$\operatorname{conv}((a_{1},-\alpha_{1}),\dots,(a_{l},-\alpha_{l}))\subset M_{\mathbb{R}}\times\mathbb{R}.$$