Gabriele Balletti: Classifications, volume bounds and universal Ehrhart inequalities of lattice polytopes

Abstract:
In this PhD thesis we study relations among invariants of lattice polytopes with particular emphasis on bounds for the volume of lattice polytopes with interior points, and inequalities for the coefficients of their Ehrhart delta polynomials. The major tools used for this investigation are explicit classifications and computer-assisted proofs. Among the most important results we give an upper bound on the dual volume of canonical Fano polytopes implying a sharp upper bound on the volume of reflexive ones, we prove the existence of "universal" inequalities in Ehrhart Theory via the generalisation of an inequality by Scott, and we present several explicit classifications of lattice polytopes in low dimensions.