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 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. Translated into toric geometry, this gives a sharp upper bound on the anti-canonical degree (−K_X)^d of a d-dimensional Fano toric variety X with at worst canonical singularities.

A well-known result by Lagarias and Ziegler states that there are finitely many equivalence classes of d-dimensional lattice polytopes having volume at most K, for fixed constants d and K. We describe an algorithm for the complete enumeration of such equivalence classes for arbitrary constants d and K. The algorithm, which gives another proof of the finiteness result, is implemented for small values of K, up to dimension six. The resulting database contains and extends several existing ones, and has been used to correct mistakes in other classifications. When specialized to three-dimensional smooth polytopes, it extends previous classifications by Bogart et al., Lorenz, and Lundman. Moreover, we give a structure theorem for smooth polytopes with few lattice points that proves that they have a quadratic triangulation and which we use, together with the classification, to describe smooth polytopes having small volume in arbitrary dimension. In dimension three we enumerate all the simplices having up to 11 interior lattice points and we use them to conjecture a set of sharp inequalities for the coefficients of the Ehrhart h∗-polynomials, unifying several existing conjectures. Finally, we extract and discuss some interesting minimal examples from the classification, and study the frequency of properties such as being spanning, very ample, IDP, and having a unimodular cover or triangulation. In particular, we find the smallest polytopes that are very ample but not IDP, and with a unimodular cover but without a unimodular triangulation.

K3 polytopes appear in complements of tropical quartic surfaces. They are dual to regular unimodular central triangulations of reflexive polytopes in the fourth dilation of the standard tetrahedron. Exploring these combinatorial objects, we classify K3 polytopes with up to 30 vertices. Their number is 36 297 333. We study the singular loci of quartic surfaces that tropicalize to K3 polytopes. These surfaces are stable in the sense of Geometric Invariant Theory.

It has been shown by Soprunov that the normalized mixed volume (minus one) of an n-tuple of n-dimensional lattice polytopes is a lower bound for the number of interior lattice points in the Minkowski sum of the polytopes. He defined n-tuples of mixed degree at most one to be exactly those for which this lower bound is attained with equality, and posed the problem of a classification of such tuples. We give a finiteness result regarding this problem in general dimension n >= 4, showing that all but finitely many n-tuples of mixed degree at most one admit a common lattice projection onto the unimodular simplex Delta(n-1). Furthermore, we give a complete solution in dimension n = 3. In the course of this we show that our finiteness result does not extend to dimension n = 3, as we describe infinite families of triples of mixed degree one not admitting a common lattice projection onto the unimodular triangle Delta(2).

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.

We associate to a finite digraph D a lattice polytope P-D whose vertices are the rows of the Laplacian matrix of D. This generalizes a construction introduced by Braun and the third author. As a consequence of the Matrix-Tree Theorem, we show that the normalized volume of P-D equals the complexity of D, and P-D contains the origin in its relative interior if and only if D is strongly connected. Interesting connections with other families of simplices are established and then used to describe reflexivity, the h*-polynomial, and the integer decomposition property of P-D in these cases. We extend Braun and Meyer's study of cycles by considering cycle digraphs. In this setting, we characterize reflexivity and show there are only four non-trivial reflexive Laplacian simplices having the integer decomposition property.

In this paper, we show the existence of universal inequalities for the h*-vector of a lattice polytope P, that is, we show that there are relations among the coefficients of the h*-polynomial that are independent of both the dimension and the degree of P. More precisely, we prove that the coefficients h* (1) and h* (2) of the h*-vector (h* (0), h* (1),..., h* (d) ) of a lattice polytope of any degree satisfy Scott's inequality if h* (3) = 0.

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 interiorlattice 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 thereare 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.

In this note we generalize and unify two results on connectivity of graphs: one by Balinsky and Barnette, one by Athanasiadis. This is done through a simple proof using commutative algebra tools. In particular we use bounds for the Castelnuovo-Mumford regularity of their Stanley-Reisner rings. As a result, if Delta is a simplicial d-pseudomanifold and s is the largest integer such that A has a missing face of size s, then the 1-skeleton of Delta is inverted right perpendicular(s)/((s+1)d)inverted left perpendicular-connected. We also show that this value is tight.