The decomposition of $\mathbb{R}^{d}$ by $\mathcal{A}^{\lambda}_{\infty}$ has the following description:

1. (1)

   The cells are labeled by the simplices $\overline{\sigma}\in S\ast\{0\}$.

2. (2)

   (The closure of) a cell $Q^{\lambda}_{\overline{\sigma}}$ is the Minkowski sum of the polytope and the cone:

   $$Q^{\lambda}_{\overline{\sigma}}=F^{\vee}_{\sigma}+\operatorname{NC}_{\Delta}(\overline{\sigma}).$$

   Here $F^{\vee}_{\sigma}$ is the face of $\partial\Delta^{\vee}_{\lambda}$ dual to $\sigma=\overline{\sigma}\cap{\partial\Delta}\in S$, (where we set $F^{\vee}_{\emptyset}=\Delta^{\vee}_{\lambda}$ for $\overline{\sigma}=\{0\}$), and $\operatorname{NC}_{\Delta}(F)$ is the normal cone to the face $F\prec\Delta$, ( in particular $\operatorname{NC}_{\Delta}(F)=\{0\}$ for $F=\Delta$). Thus, $Q^{\lambda}_{\overline{\sigma}}$ is unbounded if and only if $\overline{\sigma}=\sigma\in S$.

3. (3)

   In particular, the $d$-dimensional cells are labeled by the elements of $\operatorname{vert}(S)\cup\{0\}$. That is, there is a bounded central cell $Q^{\lambda}_{\{0\}}=\Delta^{\vee}_{\lambda}$ and unbounded cells $Q^{\lambda}_{v}$, one for each vertex $v\in\operatorname{vert}(S)$ (see Fig. 3.2).