Consider the standard simplex $\Delta^{n}$ of Example 3.65. Its indicator function induces the standard polyhedral complex in $\Delta^{n}$ consisting of the collection of its faces. The dual of $\iota_{\Delta^{n}}$, the support function $\Psi_{\Delta^{n}}$, induces a fan $\Sigma_{\Delta^{n}}:=\Pi(\Psi_{\Delta^{n}})$ of $N_{\mathbb{R}}$. The duality between these polyhedral complexes can be made explicit as