The finiteness threshold width of lattice polytopes

  title={The finiteness threshold width of lattice polytopes},
  author={M{\'o}nica Blanco and Christian Haase and Jan Hofmann and Francisco Santos},
  journal={Transactions of the American Mathematical Society, Series B},
<p>In each dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there is a constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="" alttext="w Superscript normal infinity Baseline left-parenthesis d right-parenthesis… Expand

Figures from this paper

Classification of empty lattice 4-simplices of width larger than two
<p>A lattice <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotationExpand
Complexity of linear relaxations in integer programming
For a set $X$ of integer points in a polyhedron, the smallest number of facets of any polyhedron whose set of integer points coincides with $X$ is called the relaxation complexity $\mathrm{rc}(X)$.Expand
Difference between families of weakly and strongly maximal integral lattice-free polytopes
A $d$-dimensional closed convex set $K$ in $\mathbb{R}^d$ is said to be lattice-free if the interior of $K$ is disjoint with $\mathbb{Z}^d$. We consider the following two families of lattice-freeExpand
Recognition of Digital Polyhedra with a Fixed Number of Faces Is Decidable in Dimension 3
The problem to determine whether there exists a (rational) polyhedron with at most n faces and verifying \(P \cap \mathbb {Z}^d= S\) is decidable is considered. Expand
Local optimality of Zaks-Perles-Wills simplices
  • G. Averkov
  • Mathematics, Computer Science
  • Adv. Appl. Math.
  • 2020
It is shown that S_{d,k} is a volume maximizer in the family of simplices S \in S^d(k) that have a facet with one lattice point in its relative interior that is unique up to unimodular transformations. Expand
Enumerating lattice 3-polytopes
This work presents a full enumeration of lattice 3-polytopes via their size and width, finding that most of them contain two proper subpolytopes of width larger than one, and thus can be obtained from the list of size n − 1 using computer algorithms. Expand
Classification of triples of lattice polytopes with a given mixed volume
We present an algorithm for the classification of triples of lattice polytopes with a given mixed volume $m$ in dimension 3. It is known that the classification can be reduced to the enumeration ofExpand
Classification of empty lattice 4-simplices of width larger than 2
Abstract Combining an upper bound on the volume of empty lattice 4-simplices of large width with a computer enumeration we prove the following conjecture of Haase and Ziegler (2000): Except for 179Expand
Our treatment is very combinatorial. In particular, instead of regarding a subdivision as a set of polytopes we regard it as a set of subsets of V , whose convex hulls subdivide P . This may appearExpand
The complete classification of empty lattice 4-simplices
All empty 4-simplices of width at least three are classified, including those of width two, which reveals two two-parameter families that project to the second dilation of a unimodular triangle and many more that do not. Expand


Notions of Maximality for Integral Lattice-Free Polyhedra: The Case of Dimension Three
This article considers the remaining case $d = 3$ and proves that for integral polyhedra the notions of $\mathbb{R}^3$-maximality and $\Mathbb{Z}$- maximality are equivalent. Expand
On the enumeration of lattice $3$-polytopes
A lattice $3$-polytope is a polytope $P$ with integer vertices. We call size of $P$ the number of lattice points it contains, and width of $P$ the minimum, over all integer linear functionals $f$, ofExpand
Distances Between Non-symmetric Convex Bodies and the $$MM^* $$ -estimate
AbstractLet K, D be n-dimensional convex bodes. Define the distance between K and D as $$d(K,D) = \inf \{ \lambda |TK \subset D + x \subset \lambda \cdot TK\} ,$$ where the infimum is taken overExpand
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.Expand
Projecting Lattice Polytopes Without Interior Lattice Points
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 aExpand
An Introduction to Empty Lattice Simplices
It is already NP-complete to decide whether the width of a very special class of integer simplices is 1, and it is provided for every n ≥ 3 a simple example of n-dimensional emptyinteger simplices of width n - 2, improving on earlier bounds. Expand
Enumeration of Lattice 3-Polytopes by Their Number of Lattice Points
It is proved that if P is a lattice 3-polytope of width larger than one and with at least seven lattice points then it fits in one of three categories that the authors call boxed, spiked and merged. Expand
On smooth Gorenstein polytopes
A Gorenstein polytope of index r is a lattice polytope whose rth dilate is a reflexive polytope. These objects are of interest in combinatorial commutative algebra and enumerative combinatorics, andExpand
Lattice Polytopes with Distinct Pair-Sums
It is shown that if P is a dps polytope in Rn, then N≤ 2n , and, for every n, the polytopes in RN which contain 2n lattice points are constructed. Expand
Lattice 3-Polytopes with Few Lattice Points
This work gives explicit coordinates for representatives of each class, together with other invariants such as their oriented matroid (or order type) and volume vector, and classify lattice $3$-polytopes of width larger than one and with exactly $6$ lattice points. Expand