We need a few terminologies to describe $\mathcal{A}_{\lambda,\sigma}^{\infty}$. The face of $\partial\Delta_{\lambda}^{\vee}$ dual to $\sigma$ is $F_{\sigma}^{\vee}=\partial\Delta_{\lambda}^{\vee}\cap\mathcal{A}_{\lambda,\sigma}^{\infty}$. The outward normal cone to $\sigma$ is

$$NC_{\Delta}(\sigma)=\{x\in N_{\mathbb{R}}|\langle m,x\rangle\leq\langle m^{\prime},x\rangle,\forall m\in\Delta_{\mathbb{Z}},\forall m^{\prime}\in\sigma\}.$$

By the Delzant polytope property $NC_{\Delta}(\sigma)$ is isomorphic to $\mathbb{R}_{\geq 0}^{l}$, where $n+1-l$ is the dimension of the minimal face of $\partial\Delta$ containing $\sigma$. The Minkowski sum of two sets $A,B$ means $A+B=\{a+b|a\in A,b\in B\}$.