Let notation be as in Theorem 6.6 and write $\psi=\psi_{{\overline{L}},s}$ for short. Then

(6.18)

$$\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(X_{\Sigma})=\lambda_{K}(n+1)!\int_{N_{\mathbb{R}}}(\psi^{\vee}\circ\partial\psi)\,{\mathcal{M}}_{M}(\psi),$$

where $\psi^{\vee}\circ\partial\psi$ is the integrable function defined by (3.105). When $\psi\in{\mathcal{C}}^{2}(N_{\mathbb{R}})$,

(6.19)

$$\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(X_{\Sigma})=(-1)^{n}(n+1)!\int_{N_{\mathbb{R}}}(\langle\nabla\psi(u),u\rangle-\psi(u))\det(\operatorname{Hess}(\psi))\,\,\text{\rm d}\operatorname{vol}_{N}.$$

When $\psi$ is piecewise affine,

(6.20)

$$\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(X_{\Sigma})=(n+1)!\sum_{v\in\Pi(\psi)^{0}}\int_{v^{*}}(\langle x,v\rangle-\psi(v))\,\,\text{\rm d}\operatorname{vol}_{M}(x).$$