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The normal cone of a face $F$ of a polytope consists of linear functionals whose minimum is attained on all of $F$. These cones form the inward normal fan, complete in the dual of the polytope's affine span. For a full-dimensional lattice polytope, it is a complete rational fan in the full dual space.

// Target: geometry-and-topology.bigb

For a lattice polytope $\Delta$, form the semigroup of lattice points $(m,l)$ with $l\ge 0$ and $m\in l\Delta$, using only $(0,0)$ in degree zero. Its semigroup algebra is graded by $l$, and its Proj construction gives ${\mathbb{P}}_{\Delta }$. The chart at a lattice vertex $v$ is $\mathrm{Spec}k[\mathrm{cone}(\Delta -v)\cap M]$; these charts give the normal fan of a polytope construction. For a polytope of smaller dimension, use its intrinsic affine lattice after translation.

// Target: geometry-and-topology.bigb