Three-dimensional lattice polytopes with two interior lattice points

Balletti, Gabriele; Kasprzyk, Alexander M.

doi:10.48550/arxiv.1612.08918

Cited by 6 publications

(17 citation statements)

References 21 publications

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“…Balletti and Kasprzyk [BK16] point out that hints to Conjecture 1 can also be found in older literature [Hen83,ZPW82,LZ91]. We give a short summary of the current knowledge of volume bounds for P d (k) and S d (k), with k ≥ 1.…”

Section: Introductionmentioning

confidence: 75%

“…Balletti and Kasprzyk [BK16] enumerated the family S 3 (2), up to unimodular transformations. Their enumeration allows to check that, for 59 out of 471 tetrahedra S ∈ S 3 (2), the strict inequality ν(S) > vol(S 3,2 ) holds.…”

Section: Proofsmentioning

confidence: 99%

“…Conjecture 1 was verified by complete Figure 1: . All volume maximizers in P 2 (2) [Kas09] and [BK16], respectively. The currently best upper volume bound (d(2d + 1)(s 2d+1 − 1)) d k for the whole family P d (k) is much larger than vol(S d,k ); see [AKN17].…”

Section: Introductionmentioning

confidence: 99%

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Local optimality of Zaks-Perles-Wills simplices

Averkov

2020

Advances in Applied Mathematics

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In 1982, Zaks, Perles and Wills discovered a d-dimensional lattice simplex S d,k with k interior lattice points, whose volume is linear in k and doubly exponential in the dimension d. It is conjectured that, for all d ≥ 3 and k ≥ 1, the simplex S d,k is a volume maximizer in the family P d (k) of all d-dimensional lattice polytopes with k interior lattice points. To obtain a partial confirmation of this conjecture, one can try to verify it for a subfamily of P d (k) that naturally contains S d,k as one of the members. Currently, one does not even know whether S d,k is optimal within the family S d (k) of all d-dimensional lattice simplices with k interior lattice points. In view of this, it makes sense to look at even narrower families, for example, some subfamilies of S d (k). The simplex S d,k of Zaks, Perles and Wills has a facet with only one lattice point in the relative interior. We show that S d,k is a volume maximizer in the family of simplices S ∈ S d (k) that have a facet with one lattice point in its relative interior. We also show that, in the above family, the volume maximizer is unique up to unimodular transformations.

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Section: Introductionmentioning

confidence: 75%

Section: Proofsmentioning

confidence: 99%

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Local optimality of Zaks-Perles-Wills simplices

Averkov

2020

Advances in Applied Mathematics

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“…We thank Gabriele Balletti for sharing with us his results ( [1], joint work with A. Kasprzyk) on the classification of lattice 3-polytopes with two interior points. In particular, a discrepancy between their results and a preliminary version of ours led us to correct a mistake in a first version of Theorem 7 and Corollary 1.…”

Section: Acknowledgementsmentioning

confidence: 99%

“…Tables 2 and 5 show the numbers of polytopes of each size in terms of their numbers of vertices and of interior points. Kasprzyk [12] and Balletti and Kasprzyk [1] have enumerated all lattice 3-polytopes with exactly one or two interior lattice points. Our results agree with theirs.…”

Section: Introductionmentioning

confidence: 99%

Enumeration of Lattice 3-Polytopes by Their Number of Lattice Points

Blanco

Santos

2017

Discrete Comput Geom

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We develop a procedure for the complete computational enumeration of lattice 3-polytopes of width larger than one, of which there are finitely many for each given number of lattice points. We also implement an algorithm for doing this and enumerate those with at most eleven lattice points (there are 216 453 of them).In order to achieve this we prove 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 we call boxed, spiked and merged.Boxed polytopes have at most 11 lattice points; in particular they are finitely many, and we enumerate them completely with computer help. Spiked polytopes are infinitely many but admit a quite precise description (and enumeration). Merged polytopes are computed as a union (merging) of two polytopes of width larger than one and strictly smaller number of lattice points.

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On Positivity of Ehrhart Polynomials

Liu

2019

Association for Women in Mathematics Series

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Ehrhart discovered that the function that counts the number of lattice points in dilations of an integral polytope is a polynomial. We call the coefficients of this polynomial Ehrhart coefficients, and say a polytope is Ehrhart positive if all Ehrhart coefficients are positive (which is not true for all integral polytopes). The main purpose of this article is to survey interesting families of polytopes that are known to be Ehrhart positive and discuss the reasons from which their Ehrhart positivity follows. We also include examples of polytopes that have negative Ehrhart coefficients and polytopes that are conjectured to be Ehrhart positive, as well as pose a few relevant questions.2010 Mathematics Subject Classification. Primary 52B20; Secondary 05A15, 90C57.

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Three-dimensional lattice polytopes with two interior lattice points

Cited by 6 publications

References 21 publications

Local optimality of Zaks-Perles-Wills simplices

Local optimality of Zaks-Perles-Wills simplices

Enumeration of Lattice 3-Polytopes by Their Number of Lattice Points

On Positivity of Ehrhart Polynomials

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