Hollow polytopes of large width

Codenotti, Giulia; Santos, Francisco

doi:10.1090/proc/14721

Cited by 9 publications

(10 citation statements)

References 18 publications

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“…The same is true for the usual flatness constant in the plane, which is uniquely achieved at a triangle [Hur90]. Further, the conjectured maximiser from [CS20] in three dimensions is a tetrahedron. It is thus natural to ask the following question.…”

Section: Introductionmentioning

confidence: 67%

“…Explicit values for Flt d for low dimensions are scarce: clearly, Flt 1 = 1, and Hurkens has shown that Flt 2 = 1 + 2 √ 3 [Hur90]. However, already Flt 3 is not known: in [CS20,ACMS21] the bounds 2 + √ 2 ≤ Flt 3 ≤ 3.972 are shown and it is conjectured that Flt 3 = 2 + √ 2. In [AHN19], Averkov, Hofscheier, and Nill introduced generalised flatness constants that provide a unifying approach to several questions on lattice polytopes and symplectic manifolds.…”

Section: Introductionmentioning

confidence: 99%

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Generalised Flatness Constants: A Framework Applied in Dimension $2$

Codenotti¹,

Hall²,

Hofscheier³

2021

Preprint

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Let A ∈ {Z, R} and X ⊂ R d be a bounded set. Affine transformations given by an automorphism of Z d and a translation in A d are called (affine) A-unimodular transformations. The image of X under such a transformation is called an A-unimodular copy of X. It was shown in [AHN19] that every convex body whose width is "big enough" contains an A-unimodular copy of X. The threshold when this happens is called the generalised flatness constant Flt A d (X). It resembles the classical flatness constant if A = Z and X is a lattice point. In this work, we introduce a general framework for the explicit computation of these numerical constants. The approach relies on the study of A-X-free convex bodies generalising lattice-free (also known as hollow) convex bodies. We then focus on the case that X = P is a full-dimensional polytope and show that inclusion-maximal A-P -free convex bodies are polytopes. The study of those inclusion-maximal polytopes provide us with the means to explicitly determine generalised flatness constants. We apply our approach to the case X = ∆ 2 the standard simplex in R 2 of normalised volume 1 and compute Flt R 2 (∆ 2 ) = 2 and Flt Z 2 (∆ 2 ) = 10 3 .

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Section: Introductionmentioning

confidence: 67%

Section: Introductionmentioning

confidence: 99%

Generalised Flatness Constants: A Framework Applied in Dimension $2$

Codenotti¹,

Hall²,

Hofscheier³

2021

Preprint

Self Cite

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“…Clearly, Flt(1) = 1 while in higher dimensions it is known that Flt(2) = 1 + 2 √ 3 by [28] and Flt(3) ≤ 3.972 by [3]. The most recent result known to the authors is [14,Conjecture 1.2] where it is conjectured that Flt(3) = 2 + √ 2. In [3] evidence is provided in support for this conjecture by finding a local maximizer (with respect to the Hausdorff distance), which attains the conjectured bound such that all other polytopes in a small neighbourhood have strictly smaller width.…”

Section: Convex Bodies Of Large Lattice Widthmentioning

confidence: 99%

Generalized flatness constants, spanning lattice polytopes, and the Gromov width

2021

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In this paper we motivate some new directions of research regarding the lattice width of convex bodies. We show that convex bodies of sufficiently large width contain a unimodular copy of a standard simplex. Following an argument of Eisenbrand and Shmonin, we prove that every lattice polytope contains a minimal generating set of the affine lattice spanned by its lattice points such that the number of generators (and the lattice width of their convex hull) is bounded by a constant which only depends on the dimension. We also discuss relations to recent results on spanning lattice polytopes and how our results could be viewed as the beginning of the study of generalized flatness constants. Regarding symplectic geometry, we point out how the lattice width of a Delzant polytope is related to upper and lower bounds on the Gromov width of its associated symplectic toric manifold. Throughout, we include several open questions.

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“…In this section, we would like to provide some background on the construction of the simplices in the previous section. It is inspired by the lattice-free simplices in [11] and [7], which attain the largest (known) lattice widths in dimensions 2 and 3, respectively. In fact, they can be also described in the form…”

Section: An Infinite-dimensional Viewmentioning

confidence: 99%

“…In the examples from [11,7], the lattice width is equal to v 1 − v d+1 , which is why we particularly focus on λ =…”

Section: An Infinite-dimensional Viewmentioning

confidence: 99%

Lattice-free simplices with lattice width $2d - o(d)$

Mayrhofer¹,

Jamico²,

Weltge³

2021

Preprint

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The Flatness theorem states that the maximum lattice width Flt(d) of a d-dimensional lattice-free convex set is finite. It is the key ingredient for Lenstra's algorithm for integer programming in fixed dimension, and much work has been done to obtain bounds on Flt(d). While most results have been concerned with upper bounds, only few techniques are known to obtain lower bounds. In fact, the previously best known lower bound Flt(d) ≥ 1.138d arises from direct sums of a 3-dimensional lattice-free simplex. In this work, we establish the lower bound Flt(d) ≥ 2d − O( √ d), attained by a family of lattice-free simplices. Our construction is based on a differential equation that naturally appears in this context. Additionally, we provide the first local maximizers of the lattice width of 4-and 5-dimensional lattice-free convex bodies.

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Hollow polytopes of large width

Cited by 9 publications

References 18 publications

Generalised Flatness Constants: A Framework Applied in Dimension $2$

Generalised Flatness Constants: A Framework Applied in Dimension $2$

Generalized flatness constants, spanning lattice polytopes, and the Gromov width

Lattice-free simplices with lattice width $2d - o(d)$

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