The finiteness threshold width of lattice polytopes

@article{Blanco2016TheFT,
  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},
  year={2016}
}
<p>In each dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" 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="http://www.w3.org/1998/Math/MathML" alttext="w Superscript normal infinity Baseline left-parenthesis d right-parenthesis… Expand

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References

SHOWING 1-10 OF 45 REFERENCES
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
Classification of empty lattice 4-simplices of width larger than two
A lattice $d$-simplex is the convex hull of $d+1$ affinely independent integer points in ${\mathbb R}^d$. It is called empty if it contains no lattice point apart of its $d+1$ vertices. TheExpand
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
Notions of Maximality for Integral Lattice-Free Polyhedra: The Case of Dimension Three
TLDR
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
Lattice 3-Polytopes with Few Lattice Points
TLDR
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
Enumeration of Lattice 3-Polytopes by Their Number of Lattice Points
TLDR
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
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
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
Lattice Polytopes with Distinct Pair-Sums
TLDR
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
An Introduction to Empty Lattice Simplices
TLDR
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
...
1
2
3
4
5
...