For $w\in{\partial\Delta^{\vee}}\cap\mathbb{Z}^{d}$ let $w^{\perp}$ be the set of integral points in $(\operatorname{carrier}_{\Delta^{\vee}}w)^{\vee}$, that is

$$w^{\perp}=\{m\in\Delta\cap(\mathbb{Z}^{d})^{*}\ :\ \langle m,w\rangle=1\}.$$

Then, the truncated polytope $\Delta\backslash w^{\perp}$ is defined as the convex hull of integral points of $\Delta$ which are not in $w^{\perp}$.