Classifications, volume bounds and universal Ehrhart inequalities of lattice polytopes

Abstract: In this PhD thesis we study relations among invariants of lattice polytopes. Particular emphasis is placed 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.In the first paper we give an upper bound on the volume of a polytope which is dual to a d-dimensional lattice polytope 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 paper we 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 Ehrhart delta polynomial in dimension three.In the third paper we prove the existence of inequalities for the coefficients of the Ehrhart delta polynomial of a lattice polytope P which do not depend on the degree or dimension of P. This proves that the space of all Ehrhart delta polynomials of lattice polytopes have coordinate-projections whose images do not fully cover the codomain. This is done by extending Scott's inequality to lattice polytopes whose Ehrhart delta polynomial has vanishing cubic coefficient.In the fourth paper we associate to any digraph D a simplex P whose vertices are given as the rows of the Laplacian of D, generalizing a work of Braun and Meyer. We show how basic properties of P can be read from D, for example the normalized volume of P equals the complexity of D, and P contains the origin in its relative interior if and only if D is strongly connected. We extend Braun and Meyer's study of cycles, by characterizing properties such as being Gorenstein and IDP. This is used to produce interesting examples of reflexive polytopes with non-unimodal Ehrhart delta vectors.In the fifth paper we describe an algorithm for an explicit enumeration of all equivalence classes of lattice polytopes, once dimension and volume are fixed. The algorithm is then implemented to create a database of small lattice polytopes up to dimension six. The resulting database is then compared with existing ones, used to understand the combinatorics of small smooth polytopes, and to give conjectural inequalities for coefficients of Ehrhart delta polynomials in dimension three. The frequency of some of the most important properties of lattice polytopes can be explicitly studied, and interesting minimal examples are extracted and discussed.