Let $\|\cdot\|$ be an approachable toric metric on $L^{{\text{\rm an}}}$, and denote ${\overline{L}}=(L,\|\cdot\|)$ and $\psi=\psi_{{\overline{L}},s}$ the associated concave function on $N_{\mathbb{R}}$. 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, $\pi^{\vee}_{\sigma}\colon M(\sigma)_{\mathbb{R}}\to M_{\mathbb{R}}$ the dual inclusion and $\iota\colon V(\sigma)\to X$ the closed immersion. Set $s^{\prime}=\chi^{m_{\sigma}}s$. Then

(5.76)

$$\psi_{\iota^{\ast}{\overline{L}},\iota^{\ast}s^{\prime}}=(\pi_{\sigma})_{\ast}(\psi-m_{\sigma}).$$

Dually, we have that

(5.77)

$$\psi_{\iota^{\ast}{\overline{L}},\iota^{\ast}s^{\prime}}^{\vee}=(\pi^{\vee}_{\sigma}+m_{\sigma})^{\ast}\psi^{\vee}.$$

In other words, the Legendre-Fenchel dual of $\psi_{\iota^{\ast}{\overline{L}},\iota^{\ast}s^{\prime}}$ is the restriction of $\psi^{\vee}$ to the face $F_{\sigma}$ translated by $-m_{\sigma}$.