Dual volumes and canonical forms of polytopes

A positive geometry is a semialgebraic set with an attached meromorphic form, called a canonical form, which encodes boundary information. Polytopes furnish a prototypical family of positive geometries, and their canonical forms are understood. We introduce the notion of dual (mixed) volume for polyhedra, which in particular recovers the canonical form of a polytope. Of special interest is the family of generalized permutahedra, whose canonical forms are necessary for computing certain Feynman integrals.