Size bounds and classification results for integral polyhedra with at most one interior integral point

The subject of this thesis is the analysis of integral polyhedra with at most one interior integral point. Polyhedra with no interior integral points are called lattice-free. The interest in integral lattice-free polyhedra is motivated by applications in mixed-integer optimization. Among lattice-free integral polyhedra there is a particular focus on those which are not properly contained in another lattice-free integral polyhedron. While in every given dimension, this class is finite up to unimodular transformations, a complete list of its elements was previously known only for dimensions one and two. In this thesis, a full classification of this class is given for dimension three. The second main topic of this thesis deals with integral polytopes with precisely one interior integral point. It has been known since the early 1980s that for given dimension, the volume of such a polytope is bounded from above. However, no polytopes are known for which the best known upper bounds are attained. For the special case of simplices, the sharp upper volume bound depending only on the dimension is given in this thesis. Furthermore, bounds on several other parameters describing this class of polytopes, e. g. the coefficient of asymmetry and the lattice diameter, are provided.