Consequences of the Uniform Position Principle for Spanning Lattice Polytopes

A lattice polytope is called spanning if its lattice points affinely span the ambient lattice. By comparing the spanning-assumption with other frequently used assumptions on lattice polytopes, we will see that this actually is a mild condition. On the other hand, the h^*-vector of a spanning lattice polytope satisfies strong inequalities generalizing and unifying results by Stanley and Hibi. We complete our talk by outlining the proof of these inequalities which relies on a generalization of a Uniform Position Principle statement due to Harris.