probabilities $p_2$, $p_3$, $p_6$, $p_7$, $ p_{10}$, $p_{11}$, $p_{13}$ and $p_{16}$ as free variables, the remaining eight probabilities can be expressed in terms of the free variables by using no-signaling and normalization conditions. Notice that the free variables corresponds to probabilities taking value zero in PR-box $(PR^{000})$ correlation, and for the local box $L_i$ the free variable probabilities are such that $p_k = 1$ for the correspondent free variable and zero for the other seven, where $k$ is the index of the correspondent free variable. In this way, we related $L_1\rightarrow p_2$, $L_2\rightarrow p_3$, $L_3\rightarrow p_6$, $L_4\rightarrow p_7$, $L_5\rightarrow p_{10}$, $L_6\rightarrow p_{11}$, $L_7\rightarrow p_{13}$ and $L_8\rightarrow p_{16}$.

We are interested in the region which is convex hull of the PR-box and eight local vertices $\{L_i: 1\leq i\leq 8\}$, which forms an eight dimensional simplex that we refer as the nonlocal simplex ($NLS$). In particular, we will consider the faces of $NLS$, where given a convex set $C\subseteq R^n$ and a supporting hyperplane $H$ of $C$, the set of points in $H \cap C$ defines a face of $C$ [@boyd2004convex]. Further, we will consider nonlocal faces of the region $NLS$, and all such faces can be derived by setting some of 