Dual Lattice Functions of Polytopes

Abstract

We define the dual lattice function of a rational polytope $P$ via the discrete Laplace transform of the exponential of its support function. This definition is a discrete analogue of the dual volume function of a polytope that the authors studied in previous work. We show that the dual lattice function is valuative, and by multiplying with the torus form, it becomes the canonical form of the exponential polytope $\mathrm{exp}(P)$ as a positive geometry. This result suggests the study of the class of toric polytopes, which are certain semialgebraic subsets of projective toric varieties. Our work is a first step towards discretization of positive geometries in the simplest case of polytopes.