Let $\Pi$ be a complete SCR polyhedral complex in $N_{\mathbb{R}}$ and $\psi$ an H-lattice concave function on $\Pi$. Set $\Sigma=\operatorname{rec}(\Pi)$ and $\Psi=\operatorname{rec}(\psi)$. Let $\sigma\in\Sigma$ and $m_{\sigma}\in M$ such that $\Psi|_{\sigma}=m_{\sigma}|_{\sigma}$. Let $\pi_{\sigma}\colon N_{\mathbb{R}}\to N(\sigma)_{\mathbb{R}}$ be the projection and $\pi^{\vee}_{\sigma}\colon M(\sigma)_{\mathbb{R}}\to M_{\mathbb{R}}$ the dual inclusion. Then

(4.100)

$$(\psi-m_{\sigma})(\sigma)=(\pi_{\sigma})_{\ast}(\psi-m_{\sigma}),$$

Hence the restriction of the divisor $D_{\psi-m_{\sigma}}$ to ${\mathcal{V}}(\sigma)$ corresponds to the H-lattice concave function $(\pi_{\sigma})_{\ast}(\psi-m_{\sigma})$. Dually,

(4.101)

$$(\psi-m_{\sigma})(\sigma)^{\vee}=(\pi^{\vee}_{\sigma}+m_{\sigma})^{\ast}\psi^{\vee}.$$

In other words, the Legendre-Fenchel dual of $(\psi-m_{\sigma})(\sigma)$ is the restriction of $\psi^{\vee}$ to the face $F_{\sigma}$ translated by $-m_{\sigma}$.