Let \(\sigma\) be a strictly convex polytope in \(M_\mathbb R\) (the character lattice of a torus tensored with \(\mathbb R\)). For each facet \(\tau\subset \sigma\), form the cone \[C_\tau := \cone\left(\sigma - \tau\right) = \left\{n \in M_\mathbb R ~ \middle | ~ \exists a \in \sigma, b \in \tau, \lambda \in \mathbb R_{\geq 0} \text{ s.t. } n = \lambda a - b \right\}.\]
- for a vertex \(v\), the cone \(C_v\) is obtained by translating \(\sigma\) so \(v\) is at the origin and then taking the \(\mathbb R_{\geq 0}\) span.
- for a facet \(\tau\), do the same thing but at every point in \(\tau\) and then take the union. Equivalently, you can take any point \(p\) in the relative interior of \(\tau\), translate \(\sigma\) so \(p\) is the origin, and then take the cone over that.
Note the following properties of these cones:
- since \(\sigma\) is strinctly convex, \(C_v\) does not contain a line
- the cone \(C_\tau\) contains a copy of \(\mathbb R^d\) where \(d\) is the dimension of the facet \(\tau\)
- when \(\tau \subset \tau'\) we have \(C_\tau \subset C_{\tau'}\).
The normal fan of \(\sigma\) is then \[\Sigma = \{C^\vee_{\tau} \subset N_{\mathbb R} ~ | ~ \tau \text{ is a facet of } \sigma\}.\] Equivalently, we can ignore the \(C_\tau\) for higher dimensional facets and simply set \[\Sigma = \{\tau \subset N_\mathbb R ~ | ~ \tau \preceq C_v^\vee\text{ for some vertex } v\in \sigma\}.\]