Valuations on lattice polytopes

The concept of valuation lies at the heart of geometry. A valuation is a function defined on sets that is additive with respect to unions and intersections. Volume is an example. Among the numerous further examples are surface area and more generally the intrinsic volumes as well as affine surface area and the number of points with integer coordinates in a set. Valuations arise naturally in many problems. Applications in Integral Geometry and Geometric Probability are classical. More recently, valuations have found important applications within Material Sciences, Astronomy and Tomography. A lattice polytope is the convex hull of finitely many points with integer coordinates. Such sets occur in many applications, in particular, in optimization problems and in crystallography. Within mathematics, lattice polytopes play an important role in Number Theory and as so-called Newton polytopes in Algebraic Geometry. In a recent joint paper, the principal investigators established basic results for valuations on lattice polytopes. While there has been a rapid development in the theory of valuations on convex bodies in recent years, this joint paper establishes the first new classification result for lattice polytopes since the important theorem of Betke and Kneser from 1985. The aim of the proposed research is to systematically study valuations on lattice polytopes and to obtain basic classification theorems for such valuations. Such results will find applications in Number Theory, Discrete Geometry and more applied fields.

The concept of valuation lies at the heart of geometry. A valuation is a function defined on a class of sets that is additive with respect to unions and intersections. In the project, valuations on lattice polytopes are considered. A lattice polytope is the convex hull of finitely many points with integer coordinates in n-dimensional space. Lattice polytopes are of great importance in geometry, optimization problems, discrete mathematics, and number theory, especially in the geometry of numbers. The volume and number of lattice points in a given polytope are important valuations on the space of lattice polytopes. Valuations arise naturally in many problems. Applications in integral geometry and geometric probability are classical. More recently, valuations have found important applications within materials science, astronomy and tomography. Here vector and tensor valuations on convex sets are used. This new theory was successfully extended to valuations on lattice polytopes within the research project. Specifically, the so-called reciprocity theorems of Ehrhard and Macdonald and the important classification theorem of Betke and Kneser for tensor valuations were established in important cases and contributions to Stanley's positivity theorem for tensor valuations were obtained.