A NEW CONSTRUCTION IN THE THEORY OF LATTICE COVERINGS OF AN<i>n</i>-DIMENSIONAL SPACE BY EQUAL SPHERES

Delone, B.; Долбилин, Николай Петрович; Ryškov, S S; Štogrin, M I

doi:10.1070/im1970v004n02abeh000904

Cited by 12 publications

(12 citation statements)

References 4 publications

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“…Their investigations were continued by the author [25]. Lattice coverings with balls were investigated by Barnes and Dickson [4], Delone et al [12], Schürmann and Vallentin [41] and the author [21]. For uniqueness problems of lattice packings and coverings of extreme density, see the author [23].…”

Section: Introductionmentioning

confidence: 98%

Application of an idea of Voronoĭ to lattice zeta functions

Gruber

2012

Proc. Steklov Inst. Math.

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A major problem in the geometry of numbers is the investigation of the local minima of the Epstein zeta function. In this article refined minimum properties of the Epstein zeta function and more general lattice zeta functions are studied. Using an idea of Voronoȋ, characterizations and sufficient conditions are given for lattices at which the Epstein zeta function is stationary or quadratic minimum. Similar problems of a duality character are investigated for the product of the Epstein zeta function of a lattice and the Epstein zeta function of the polar lattice. Besides Voronoȋ type notions such as versions of perfection and eutaxy, these results involve spherical designs and automorphism groups of lattices. Several results are extended to more general lattice zeta functions, where the Euclidean norm is replaced by a smooth norm.

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Section: Introductionmentioning

confidence: 98%

Application of an idea of Voronoĭ to lattice zeta functions

Gruber

2012

Proc. Steklov Inst. Math.

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“…The combined work of Barnes and Dickson [4], Delone, Dolbilin, Ryshkov, and Stogrin [14], and Schürmann and Vallentin [64] yields a characterization of lattice coverings of balls which have locally minimum density (see also the author's paper [32]). Characterizations of lattice coverings of minimum density have been proved so far only for Euclidean balls, while corresponding packing results have also been proved for convex bodies (see Section 3).…”

Section: Local Minima Of the Lattice Covering Density Of Ballsmentioning

confidence: 96%

“…In the context of projective geometry, earlier similar ideas are due to Plücker [49] and Klein [40]. The idea of Voronoȋ has been applied successfully to refinements of Voronoȋ's criterion (see [5,73,30]), lattice coverings with balls of extreme density (see [4,14,64,32]), minima of the Epstein zeta function (see [16,52,12,57,31]), geodesics on Riemannian manifolds of a Teichmüller space (see [59][60][61][62]), and John type and minimum position problems (see [23,27,35,33]). …”

Section: Local Maxima Of the Lattice Packing Densitymentioning

confidence: 98%

Lattice packing and covering of convex bodies

Gruber

2011

Proc. Steklov Inst. Math.

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The aim of this article is twofold. First, to indicate briefly major problems and developments dealing with lattice packings and coverings of balls and convex bodies. Second, to survey more recent results on uniqueness of lattice packings and coverings of extreme density, on characterization of local minima and maxima of the density and on estimates of the kissing number. Emphasis is on results in general dimensions.

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“…Ryshkov and Delone [DR63] solved the lattice covering problem in dimension 4 by this method. However, the five-dimensional lattice covering problem cannot be solved in this way.…”

Section: Heuristic Methodsmentioning

confidence: 99%

“…The PQF Q = (q ij ) defines the inner product of a Euclidean space (R d , (·, ·)) by (x, y) = x t Qy. From the article [DDRS70], §3, of Delone, Dolbilin, Ryshkov and Stogrin we can extract the following proposition which will be central in our further discussion.…”

Section: An Lmi Constraint For the Inhomogeneous Minimummentioning

confidence: 99%

Computational Approaches to Lattice Packing and Covering Problems

Schürmann

Vallentin

2005

Discrete Comput Geom

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We describe algorithms which address two classical problems in lattice geometry: the lattice covering and the simultaneous lattice packing-covering problem. Theoretically our algorithms solve the two problems in any fixed dimension d in the sense that they approximate optimal covering lattices and optimal packing-covering lattices within any desired accuracy. Both algorithms involve semidefinite programming and are based on Voronoi's reduction theory for positive definite quadratic forms, which describes all possible Delone triangulations of Z d . In practice, our implementations reproduce known results in dimensions d ≤ 5 and in particular solve the two problems in these dimensions. For d = 6 our computations produce new best known covering as well as packing-covering lattices, which are closely related to the lattice E * 6 . For d = 7, 8 our approach leads to new best known covering lattices. Although we use numerical methods, we made some effort to transform numerical evidences into rigorous proofs. We provide rigorous error bounds and prove that some of the new lattices are locally optimal.

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A NEW CONSTRUCTION IN THE THEORY OF LATTICE COVERINGS OF ANn-DIMENSIONAL SPACE BY EQUAL SPHERES

Cited by 12 publications

References 4 publications

Application of an idea of Voronoĭ to lattice zeta functions

Application of an idea of Voronoĭ to lattice zeta functions

Lattice packing and covering of convex bodies

Computational Approaches to Lattice Packing and Covering Problems

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