The previous example can be generalized to an arbitrary polytope $\Delta\subset M_{\mathbb{R}}$ . The indicator function $\iota_{\Delta}$ induces the standard decomposition of $\Delta$ into its faces and dually, the support function $\Psi_{\Delta}$ induces a polyhedral complex $\Sigma_{\Delta}:=\Pi(\Psi_{\Delta})$ made of cones. If $\Delta$ is of maximal dimension, then $\Sigma_{\Delta}$ is a fan.

The faces of $\Delta$ are in one-to-one correspondence with the cones of $\Sigma_{\Delta}$ through the Legendre-Fenchel correspondence. For a face $F$ of $\Delta$, its corresponding cone is

$$\sigma_{F}:=F^{*}=\{u\in N_{\mathbb{R}}\mid\langle u,x-y\rangle\geq 0\mbox{ for all }x\in\Delta,y\in F\}.$$

Reciprocally, to each cone $\sigma$ corresponds a face of $\Delta$ of complementary dimension

$$F_{\sigma}:=\sigma^{*}=\{x\in\Delta\mid\langle x,u\rangle=\Psi_{\Delta}(u)\mbox{ for all }u\in\sigma\}.$$

On a cone $\sigma\in\Sigma$, the function $\Psi_{\Delta}$ is defined by any vector $m_{\sigma}$ in the affine space $\operatorname{aff}(F_{\sigma})$. The cone $\sigma$ is normal to $F_{\sigma}$.