Maximal Lattice-free Convex sets in 3 Dimensions

Given a lattice Λ, a lattice polyhedra is a convex polyhedra P , such that all vertices of P are lattice points and no other other point in P is a lattice point. By lattice-free convex set we mean a convex set with no lattice point from Λ in its strict interior. However lattice points are allowed on the boundary. We will usually work with the standard lattice in R3, i.e. points which have all three coordinates integral. We study maximal lattice-free convex sets in three dimensions in this paper. We now define some three dimensional polyhedra that we will need. A tetrahedron is a simplex, i.e. an affine transformation of the convex hull of {(0,0,0), (1,0,0), (0,1,0), (0,0,1)}. Given a two dimensional convex polygon P and a vector d in R3, a cylinder over P in the direction d is the polyhedron {x + γd : x ∈ P, γ ∈ R }. A cone C is pointed if C ∩−C = {0}. An unbounded pyramid is a polyhedron of the form {v + C}, where v is a point in R3 and C is a pointed cone generated by 4 distinct rays in R3. A valley is the intersection of two half-spaces a1 · x ≤ 1, a2 · x ≤ 1. Note that a valley might be empty, or be one of the original halfspaces or be equivalent to a split if a1, a2 are not linearly independent. (Note that, in this paper, a split is the region between two parallel ∗Supported by a Mellon Fellowship. †Supported by NSF grant CMMI0653419, ONR grant N00014-97-1-0196 and ANR grant BLAN06-1-138894. ‡Supported by ONR grant N00014-97-1-0196.