An Introduction to Empty Lattice Simplices

We study simplices whose vertices lie on a lattice and have no other lattice points. Suchèmpty lattice simplices' come up in the theory of integer programming, and in some combi-natorial problems. They have been investigated in various contexts and under varying terminology Can thèemptiness' of lattice simplices bèwell-characterized' ? Is theirìattice-width' small ? Do the integer points of the parallelepiped they generate have a particular structure ? Thègood characterization' of empty lattice simplices occurs to be open in general ! We provide a polynomial algorithm for deciding when a given integer`knapsack' or`partition' lattice simplex is empty. More generally, we ask for a characterization of linear inequalities satissed by the lattice points of a lattice parallelepiped. We state a conjecture about such inequalities, prove it for n 4, and deduce several variants of classical results of Reeve, White and Scarf characterizing the emptiness of small dimensional lattice simplices. For instance, a three dimensional integer simplex is empty if and only if all its faces have width 1. Seemingly diierent characterizations can be easily proved from one another using the Hermite normal form. In xed dimension the width of polytopes can be computed in polynomial time (see the simple integer programming formulation of Haase and Ziegler). We prove that it is already NP-complete to decide whether the width of a very special class of integer simplices is 1, and we also provide for every n 3 a simple example of n-dimensional empty integer simplices of width n ? 2, improving on earlier bounds.