On conjectures of Minkowski and Woods for n=8

Hans-Gill, R. J.; Raka, Madhu; Sehmi, Ranjeet

doi:10.4064/aa147-4-3

Cited by 22 publications

(39 citation statements)

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“…As a consequence of the work of this paper we have that Minkowski's conjecture is correct in dimension 7, thereby confirming [29].…”

Section: ]supporting

confidence: 80%

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The Seven Dimensional Perfect Delaunay Polytopes and Delaunay Simplices

Sikirić¹

2017

Can. j. math.

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Abstract. For a lattice L of R n , a sphere S(c, r) of center c and radius r is called empty if for any v ∈ L we have v − c ≥ r. Then the set S(c, r) ∩ L is the vertex set of a Delaunay polytope P = conv(S(c, r) ∩ L). A Delaunay polytope is called perfect if any affine transformation φ such that φ(P ) is a Delaunay polytope is necessarily an isometry of the space composed with an homothety.Perfect Delaunay polytopes are remarkable structure that exist only if n = 1 or n ≥ 6 and they have shown up recently in covering maxima studies. Here we give a general algorithm for their enumeration that relies on the Erdahl cone. We apply this algorithm in dimension 7 which allow us to find that there are only two perfect Delaunay polytopes: 3 21 which is a Delaunay polytope in the root lattice E 7 and the Erdahl Rybnikov polytope.We then use this classification in order to get the list of all types Delaunay simplices in dimension 7 and found 11 types.

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“…As a consequence of the work of this paper we have that Minkowski's conjecture is correct in dimension 7, thereby confirming [29].…”

Section: ]supporting

confidence: 80%

“…Recently, it has been proved for n ≤ 8 in [31,32,30,29] by computational methods based on Korkine-Zolotarev reduction theory. Other theoretical approaches have been attempted in [37,42] by Dynamical System Theory.…”

Section: Introductionmentioning

confidence: 99%

The Seven Dimensional Perfect Delaunay Polytopes and Delaunay Simplices

Sikirić¹

2017

Can. j. math.

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“…For n = 4, the conjecture was proved by Dyson [13], Bambah and Woods [2] and Skubenko [30]. Following this approach which has come to be known as Remak-Davenport approach, Minkowski's Conjecture has been further proved for 5 ≤ n ≤ 8 by the efforts of various mathematicians : Skubenko [31], Bambah and Woods [3], McMullen [23], Woods [35,36] and authors [18,19]. For a detailed history of Minkowski's conjecture and related results, see Gruber [16], Gruber and Lekkerkerker [15] and Bambah et al [4].…”

Section: Introductionmentioning

confidence: 96%

Estimates on Conjectures of Minkowski and woods

Hans-Gill

Raka

Sehmi

2010

Indian J Pure Appl Math

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Let R n be the n-dimensional Euclidean space. Let ∧ be a lattice of determinant 1 such that there is a sphere |X| < R which contains no point of ∧ other than the origin O and has n linearly independent points of ∧ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in R n of radius n/4 contains a point of ∧. This is known to be true for n ≤ 8. Here we give estimates on a more general conjecture of Woods for n ≥ 9. This leads to an improvement for 9 ≤ n ≤ 22 on estimates of Il'in (1991) to the long standing conjecture of Minkowski on product of n non-homogeneous linear forms.

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“…This conjecture is proved for n ≤ 8, cf. [13], [9], [10]. Several recent results concern the case of abelian fields.…”

Section: Introductionmentioning

confidence: 99%