#### Author of Thesis:

Gabriele Balletti

#### Title of Thesis:

Classification and volume bounds of lattice polytopes

#### External reviewer:

Martin Henk (TU Berlin)

#### Supervisor:

Benjamin Nill

#### Time:

Wednesday, March 8, 2017, at. 3:15 p.m.

#### Place:

Room 14, house 5, Kräftriket

#### Abstract:

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.