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.