On packing of Minkowski balls. II (original) (raw)
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On Lattice points and Optimal Packing of Minkowski's Balls and Domains
arXiv (Cornell University), 2023
We investigate lattice packings of Minkowski's balls and domains, as well as the distribution of lattice points on Minkowski's curves which are boundaries of Minkowski's balls. By results of the proof of Minkowski's conjecture about the critical determinant we devide the balls and domains on 3 classes: Minkowski, Davis and Chebyshev-Cohn balls. The optimal lattice packings of the balls and domains are obtained. The minimum areas of hexagons inscribed in the balls and domains and circumscribed around their are given. We construct direct systems of these balls, domains and their critical lattices and calculate their direct limits.
Proceedings of the Bulgarian Academy of Sciences
We investigate lattice packings of Minkowski balls. By the results of the proof of Minkowski conjecture about the critical determinant we divide Minkowski balls into 3 classes: Minkowski balls, Davis balls and Chebyshev–Cohn balls. We investigate lattice packings of these balls on planes with varying Minkowski metric and search among these packings the optimal packings. In this paper we prove that the optimal lattice packing of the Minkowski, Davis, and Chebyshev–Cohn balls is realized with respect to the sublattices of index two of the critical lattices of corresponding balls.
Minkowski Arrangements of Spheres
Monatshefte f�r Mathematik, 2004
Let " be a non-negative number not greater than 1. Consider an arrangement S of (not necessarily congruent) spheres with positive homogenity in the n-dimensional Euclidean space, i.e., in which the infimum of the radii of the spheres divided by the supremum of the radii of the spheres is a positive number. With each sphere S of S associate a concentric sphere of radius " times the radius of S. We call this sphere the "-kernel of S. The arrangement S is said to be a Minkowski arrangement of order " if no sphere of S overlaps the "-kernel of another sphere. The problem is to find the greatest possible density d n ð"Þ of n-dimensional Minkowski sphere arrangements of order ". In this paper we give upper bounds on d n ð"Þ for " 4 1 n. 2000 Mathematics Subject Classification: 52C17
Further lattice packings in high dimensions
Mathematika, 1982
Barnes and Sloane recently described a “general construction” for lattice packings of equal spheres in Euclidean space. In the present paper we simplify and further generalize their construction, and make it suitable for iteration. As a result we obtain lattice packings in ℝm with density Δ satisfying , as m → ∞ where is the smallest value of k for which the k-th iterated logarithm of m is less than 1. These appear to be the densest lattices that have been explicitly constructed in high-dimensional space. New records are also established in a number of lower dimensions, beginning in dimension 96.
Self-packing of centrally symmetric convex bodies in ℝ2
Discrete & Computational Geometry, 1992
Let B be a compact convex body symmetric around 0 in R 2 which has nonempty interior, i.e., the unit ball of a two-dimensional Minkowski space. The self-packing radius p(m, B) is the smallest t such that tB can be packed with m translates of the interior of B. For m < 6 we show that the self-packing radius p(m, B) = 1 + 2#t(m, B) where ~t(m, B) is the Minkowski length of the side of the largest equilateral m-gon inscribed in B (measured in the Minkowski metric determined by B). We show p(6, B) = p(7, B) = 3 for all B, and determine most of the largest and smallest values of p(m, B) for m < 7. For all m we have 6(B)/-2 < p(m, B) < \~j/ + 1, where 3(B) is the packing density of B in •2
Packings by translation balls in S̃L 2 ( R )
2014
For one of Thurston model spaces, S̃L2(R), we discuss translation balls and packing that space by such balls in contrast to the packing by standard (geodesic) balls. We present an infinite family of packings generated by discrete groups of isometries, and observe numerical results on their densities. In particular, we found packings whose densities are close to the upper bound density for ball packings in the hyperbolic 3-space. Mathematics Subject Classification (2000). 52C17, 52C22, 52B15, 53A35, 51M20. Co-authors are: Emil Molnár, jenő Szirmai and Andrei Vesnin
Generalized Apollonian packings
Communications in Mathematical Physics, 1990
In this paper we generalize the classical two-dimensional Apollonian packing of circles to the case where the circles are no more tangent. We introduce two elements of SL(2, <C) as generators: R and T that are hyperbolic rotations of -and -(JV = 2,3,4,...), around two distinct points. The limit set of the discrete 3 N group generated by R and T provides, for N = 7,8,9,... a generalization of the Apollonian packing (which is itself recovered for JV = oo). The values JV = 2,3,4,5 produce a very different result, giving rise to the rotation groups of the cube for JV = 2 and 4, and the icosahedron for JV = 3 and 5. For JV = 6 the group is no longer discrete. To further analyze this structure for JV ^ 7, we move to the Minkowski space in which the group acts on a one sheeted hyperboloid. The circles are now represented by points on this variety and generate a crystal on it.
New Lower Bound for the Optimal Ball Packing Density in Hyperbolic 4-Space
Discrete & Computational Geometry, 2014
In this paper we consider ball packings of 4-dimensional hyperbolic space, and show that it is possible to exceed the conjectured 4-dimensional realizable packing density upper bound due to L. Fejes-Tóth (Regular Figures, 1964). We give seven examples of horoball packing configurations that yield higher densities of ≈ 0.71644896 where horoballs are centered at ideal vertices of Coxeter simplices that make up fundamental domains of certain Coxeter simplex reflection groups. * Mathematics Subject Classification 2010: 52C17, 52C22, 52B15.
On Lattice Packings and Coverings of Asymmetric Limited-Magnitude Balls
IEEE Transactions on Information Theory, 2021
We construct integer error-correcting codes and covering codes for the limited-magnitude error channel with more than one error. The codes are lattices that pack or cover the space with the appropriate error ball. Some of the constructions attain an asymptotic packing/covering density that is constant. The results are obtained via various methods, including the use of codes in the Hamming metric, modular Bt-sequences, 2-fold Sidon sets, and sets avoiding arithmetic progression.