Dual cone (${\sigma }^{\vee }$)

For a cone $\sigma \subseteq N_{\mathbb{R}}$, its dual cone is

For a closed polyhedral cone $C$, the maps $\tau \mapsto C^{\vee }\cap {\tau }^{\perp }$ and $\nu \mapsto C\cap {\nu }^{\perp }$ give inverse order-reversing bijections on faces. Choose a exposing functional in the relative interior of the dual face. The bipolar identity and an exposing functional for the original face show that applying the maps twice returns that face.

// Target: geometry-and-topology.bigb

For a rational cone $\sigma$, the lattice points ${\sigma }^{\vee }\cap M$ form a finitely generated additive monoid.

The lattice points of a rational polyhedral cone form a finitely generated additive monoid. Choose lattice generators for the cone. Subtracting integer parts of nonnegative coefficients leaves a lattice point in their bounded fundamental parallelepiped; its finitely many lattice points, together with the chosen generators, generate the monoid. This supplies finite generation of the coordinate ring of an affine toric variety.

// Target: geometry-and-topology.bigb

The Hilbert basis is the unique minimal generating set of the additive monoid of lattice points in a pointed rational cone.