On the Number of Lattice Free Polytopes

We show that up to unimodular equivalence in each dimension there are only finitely many lattice polytopes without interior lattice points that do not admit a lattice projection onto a lower-dimensional lattice polytope without interior lattice points. This was conjectured by Treutlein [Treutlein, J. 2008. 3-Dimensional lattice polytopes without interior lattice points. September 10, http://arXiv.org/abs/0809.1787.] As an immediate corollary, we get a short proof of a recent result of Averkov, Wagner, and Weismantel [Averkov, G., C. Wagner, R. Weismantel. 2010. Maximal lattice-free polyhedra: Finiteness and an explicit description in dimension three. Math. Oper. Res. Forthcoming.], namely, the finiteness of the number of maximal lattice polytopes without interior lattice points. Moreover, we show that, in dimension four and higher, some of these finitely many polytopes are not maximal as convex bodies without interior lattice points.

show abstract

“…Recently, in Averkov et al [2], it was proven that 3 = 12. Therefore, a positive answer to Question 1.1 would finish the classification of 3 .…”

Section: The Main Theoremmentioning

confidence: 96%

Projecting Lattice Polytopes Without Interior Lattice Points

Nill

Ziegler

2011

Mathematics of OR

View full text Add to dashboard Cite

show abstract

“…We will show that this bound is also sharp when both d and k grow large by exhibiting an extensive family of d-dimensional lattice polytopes contained in [0, k] d such that every lattice point in [0, k] d can either be inserted in or deleted from these polytopes. We first prove the following about Λ (2). Thereafter, by a unit square we mean the square [0, 1] 2 or any of its translates by a lattice vector.…”

Section: The Number Of Possible Insertion and Deletion Movesmentioning

confidence: 99%

Elementary moves on lattice polytopes

David

Pournin

Rakotonarivo

2020

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

We introduce a graph structure on Euclidean polytopes. The vertices of this graph are the d-dimensional polytopes contained in R d and its edges connect any two polytopes that can be obtained from one another by either inserting or deleting a vertex, while keeping their vertex sets otherwise unaffected. We prove several results on the connectivity of this graph, and on a number of its subgraphs. We are especially interested in several families of subgraphs induced by lattice polytopes, such as the subgraphs induced by the lattice polytopes with n or n + 1 vertices, that turn out to exhibit intriguing properties.

show abstract

“…This will not be proved here. (Polytopes with the latter property are investigated by Bárány, Howe, and Scarf [3], Bárány and Kantor [4], Bárány, Scarf and Shallcross [5], Deza and Onn [9], etc.) 8.…”

Section: Background On Semiordersmentioning

confidence: 99%

The Representation Polyhedron of a Semiorder

Balof

Doignon

Fiorini

2011

Order

View full text Add to dashboard Cite

Abstract. Let a finite semiorder, or unit interval order, be given. All its numerical representations (when suitably defined) form a convex polyhedron. We show that the facets of the representation polyhedron correspond to the noses and hollows of the semiorder. Our main result is to prove that the coordinates of the vertices and the components of the extreme rays of the polyhedron are all integral multiples of a common value. The result follows from the total dual integrality of the system defining the representation polyhedron. Total dual integrality is in turn derived from a particular property of the oriented cycles in the directed graph of noses and hollows of a strictly upper diagonal step tableau. Our approach delivers also a new proof for the existence of the minimal representation, a concept originally discovered by Pirlot (1990). Finding combinatorial interpretations of the vertices and extreme rays of the representation polyhedron is left for future work.

show abstract

On the Number of Lattice Free Polytopes

Cited by 7 publications

References 16 publications

Projecting Lattice Polytopes Without Interior Lattice Points

Projecting Lattice Polytopes Without Interior Lattice Points

Elementary moves on lattice polytopes

The Representation Polyhedron of a Semiorder

Contact Info

Product

Resources

About