Abstract:

Lattice-point-free convex bodies have been studied intensively in the geometry of numbers. In the world of lattice polytopes their counterparts are hollow lattice polytopes: lattice polytopes without interior lattice points. There are many open questions and conjectures about these interesting geometric objects.

In this talk I will discuss two invariants, the degree and the codegree, that appear naturally in Ehrhart theory and that give a quantitative measure of the 'hollowness' of a lattice polytope. The main result I would like to present is a structure theorem on lattice polytopes with large codegree. Its main application is a finiteness result on Ehrhart h*-polynomials.

This is joint work with Victor Batyrev, Christian Haase and Sam Payne.