Let $\Sigma$ be a complete fan in $N_{\mathbb{R}}$ and $\Psi\colon N_{\mathbb{R}}\to\mathbb{R}$ a support function on $\Sigma$.

1. (1)

   Let $\sigma\in\Sigma$, $F_{\sigma}$ the associated face of $\Delta_{\Psi}$, and $m_{\sigma}^{\prime}\in F_{\sigma}\cap M$. Let $\pi_{\sigma}\colon N_{\mathbb{R}}\to N(\sigma)_{\mathbb{R}}$ be the natural projection. Then

   (4.48)

   $$(\Psi-m^{\prime}_{\sigma})(\sigma)=(\pi_{\sigma})_{\ast}(\Psi-m_{\sigma}^{\prime}).$$

   In particular, the restriction of $D_{\Psi-m^{\prime}_{\sigma}}$ to $V(\sigma)$ is given by the concave function $(\pi_{\sigma})_{\ast}(\Psi-m_{\sigma}^{\prime})$. Moreover, the associated polytope is

   (4.49)

   $$\Delta_{(\Psi-m_{\sigma}^{\prime})(\sigma)}=F_{\sigma}-m_{\sigma}^{\prime}\subset M(\sigma)_{\mathbb{R}}=\sigma^{\bot}.$$

2. (2)

   Let $H\colon N^{\prime}\to N$ be a linear map and $H^{\vee}\colon M\to M^{\prime}$ its dual map, where $M^{\prime}=(N^{\prime})^{\vee}$. Let $\Sigma^{\prime}$ be a fan in $N^{\prime}_{\mathbb{R}}$ such that, for each $\sigma^{\prime}\in\Sigma^{\prime}$ there is $\sigma\in\Sigma$ with $H(\sigma^{\prime})\subset\sigma$, and let $p\in X_{\Sigma,0}(K)$. Then

   (4.50)

   $$\varphi_{p,H}^{\ast}D_{\Psi}=D_{H^{\ast}\Psi},$$

   and the associated polytope is

   (4.51)

   $$\Delta_{H^{\ast}\Psi}=H^{\vee}(\Delta_{\Psi})\subset M^{\prime}_{\mathbb{R}}.$$