All statements follow easily from the definition of $L_{\lambda}$. Namely, the cells correspond to the subsets $I\subset\Delta\cap(\mathbb{Z}^{d})^{*}$: the corresponding linear functions $\langle m,n\rangle+\lambda(m),\ m\in I$, saturate the maximum in $L_{\lambda}$. Since $\lambda$ is a concave function this can happen only if $I$ is a set of vertices of some simplex $\overline{\sigma}\in S\ast\{0\}$. This proves (1).

For (3) we notice that a $d$-cell is a domain of linearity of $L_{\lambda}$, labeled by the vertex $m\in\operatorname{vert}(S)\cup\{0\}$ whose corresponding linear function $\langle m,n\rangle+\lambda(m)$ is maximal. In particular, the central cell $Q^{\lambda}_{\{0\}}$ is the set of $n\in\mathbb{R}^{d}$, such that the maximum is achieved by $\langle\{0\},n\rangle+\lambda(0)$, i.e.

which are exactly the defining inequalities for $\Delta^{\vee}_{\lambda}$.

More generally, a point $n$ is in (the closure of) $Q^{\lambda}_{\overline{\sigma}}$ if and only if:

which are exactly the defining inequalities for the polyhedron $F^{\vee}_{\sigma}+\operatorname{NC}_{\Delta}(\overline{\sigma})$. ∎