A polyhedron defined by an integral valued constraint matrix and an integral valued right-hand side is lattice-free if it does not contain an element of the integer lattice. In this talk, we present a link between the lattice-freeness of polyhedra, the diameter of finite abelian groups and the height of Hilbert bases. As a result, we will be able to prove novel upper bounds on the lattice width of lattice-free pyramids if a conjecture regarding the height of Hilbert bases holds. Further, we improve existing lattice width bounds of lattice-free simplices. All our bounds are independent of the dimension and solely depend on the maximal minors of the constraint matrix.
The second part of the talk is devoted to a study of the above-mentioned conjecture. We completely characterize the Hilbert basis of a pointed polyhedral cone when all the maximal minors of the constraint matrix are bounded by two in absolute value. This can be interpreted as an extension of a well-known result which states that the Hilbert basis elements lie on the extreme rays if the constraint matrix is unimodular, i.e., all maximal minors are bounded by one in absolute value.
This is joint work with Martin Henk and Robert Weismantel.