Maximal Lattice-Free Polyhedra: Finiteness and an Explicit Description in Dimension Three

Abstract

A convex set with nonempty interior is maximal lattice-free if it is inclusion-maximal with respect to the property of not containing integer points in its interior. Maximal lattice-free convex sets are known to be polyhedra. The precision of a rational polyhedron P in R d is the smallest natural number s such that sP is an integral polyhedron. In this paper we show that, up to affine mappings preserving Z d , the number of maximal lattice-free rational polyhedra of a given precision s is finite. Furthermore, we present the complete list of all maximal lattice-free integral polyhedra in dimension three. Our results are motivated by recent research on cutting plane theory in mixed-integer linear optimization.2010 MSC: 52B20, 52C07, 90C10, 90C11

Notions of Maximality for Integral Lattice-Free Polyhedra: The Case of Dimension Three

“…It turns out that every integral lattice-free polyhedron is a subset of a Z d -maximal integral lattice-free polyhedron. This is stated without proof in [DW12] and can be proven using results in [NZ11] or arguments from [AWW11]. Consequently, the family of all Z d -maximal integral lattice-free polyhedra has finite convergence property for every n. Furthermore, no proper subfamily of the latter family has finite convergence when n > 0, as follows from results of [Del12].…”

“…Recently, the family of Z d -maximal integral lattice-free polyhedra has attracted attention of experts in algebraic geometry and optimization; see [Tre08,Tre10], [NZ11] and [AWW11]. With a view towards applications in the mentioned research areas, having a better geometric understanding of such polyhedra is desirable.…”

“…In contrast, so far no simple description of Z d -maximality is available, even in the case of integral polyhedra. This has motivated the following question, asked in [NZ11] and [AWW11]: is every Z dmaximal integral lattice-free polyhedron also R d -maximal? This can also be formulated as follows: is it true for every integral lattice-free polyhedron P , that if conv(P ∪ {x}) is lattice-free for some x ∈ R d \ P , then conv(P ∪ {x}) is lattice-free for some x ∈ Z d \ P ?…”

“…For dimensions d = 1 and d = 2 enumeration can be carried out easily. Already, for dimension d = 3, this is a challenging problem which was only partially addressed in [AWW11]. Using properties of R d -maximal lattice-free sets, the authors of [AWW11] were able to enumerate the family of all R 3 -maximal integral lattice-free polyhedra.…”

“…In this article, we consider the remaining case d = 3 and prove that for integral polyhedra the notions of R 3 -maximality and Z 3 -maximality are equivalent. As a consequence, the classification of all R 3 -maximal integral polyhedra by Averkov, Wagner and Weismantel (2011) contains all Z 3 -maximal integral polyhedra. …”