Gabriele Balletti: Classification and volume bounds of lattice polytopes

In this licentiate thesis we study relations among invariants of lattice polytopes, with particular focus on bounds for the volume. In the first paper we give an upper bound on the volume vol(\(P^*\)) of a polytope \(P^*\) dual to a d-dimensional lattice polytope P with exactly one interior lattice point, in each dimension d. This bound, expressed in terms of the Sylvester sequence, is sharp, and is achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d-dimensional reflexive polytope. In the second paperwe classify the three-dimensional lattice polytopes with two lattice points 
in their strict interior. Up to unimodular equivalence there are 22,673,449 such polytopes. This classification allows us to verify, for this case only, the sharp conjectural upper bound for the volume of a lattice polytope with interior points, and provides strong evidence for more general new inequalities on the coefficients of the \(h^*\)-polynomial in dimension three.