Packing and Covering with Convex Sets

Abstract. We prove that if a (th r )-convex domain in the hyperbolic plane is covered by n ≥ 2 circular discs of radius r , then the density of the covering is larger than 2π/ √ 27. The density bound is optimal, and the condition of (th r )-convexity is essentially optimal. Combining our result with earlier estimates yields that if at least two non-overlapping equal circular discs cover a given circular disc in a surface of constant curvature, then the density of the covering is larger than 2π/ √ 27.

“…are positive for 0 < s ≤ 1/3.5, and both inequalities (12) and (13) hold for ch r = 3.5, we conclude (9).…”

Section: Proof Of Theorem 11supporting

confidence: 53%

“…Therefore we have a stronger bound for finite coverings of (th r )-convex domains (see Theorem 1.1) than for infinite coverings. For further related results, see the surveys [12] and [15].…”

Section: Corollary 14 If a (Th R )-Convex Domain In Hmentioning

confidence: 99%

Finite Coverings in the Hyperbolic Plane

Böröczky

2004

“…For additional related results and references on this topic, see [1], [2], [20], [22], and the survey papers [4] and [5].…”

Section: Moreover the Arrangement Of 18 Mutually Nonoverlapping Tranmentioning

confidence: 99%

“…Thus Proposition 4.3 is not so closely connected to the other propositions and lemmas, but it has an interest of its own. We recall some notions from the theory of packings (see [4] and [5]). …”

Section: Applicationsmentioning

confidence: 99%

The Translative Kissing Number of Tetrahedra Is 18

Talata

1999

We show that the maximum number of mutually nonoverlapping translates of any tetrahedron T which touch T is 18. Moreover, in the case of 18 touching translates the arrangement turns out to be unique. We also give a description of all possible arrangements of 17 touching translates. Finally, we apply these results to determine the minimum and maximum densities of 17 + -neighbor translative packings of tetrahedra.

“…In connection with this we briefly mention some of the exciting results of lower dimensions. The reader is referred to [10], [12], [22] and [25] for a comprehensive view of results of this type.…”

Section: Introductionmentioning

confidence: 99%

Improving Rogers’ Upper Bound for the Density of Unit Ball Packings via Estimating the Surface Area of Voronoi Cells from Below in Euclidean \sl d -Space for All \sl d ≥ \bf 8

Bezdek¹

2002

The sphere packing problem asks for the densest packing of unit balls in E d . This problem has its roots in geometry, number theory and information theory and it is part of Hilbert's 18th problem. One of the most attractive results on the sphere packing problem was proved by Rogers in 1958. It can be phrased as follows. Take a regular d-dimensional simplex of edge length 2 in E d and then draw a d-dimensional unit ball around each vertex of the simplex. Let σ d denote the ratio of the volume of the portion of the simplex covered by balls to the volume of the simplex. Then the volume of any Voronoi cell in a packing of unit balls in E d is at least ω d /σ d , where ω d denotes the volume of a d-dimensional unit ball. This has the immediate corollary that the density of any unit ball packing in E d is at most σ d . In 1978 Kabatjanskii and Levenštein improved this bound for large d. In fact, Rogers' bound is the presently known best bound for 4 ≤ d ≤ 42, and above that the Kabatjanskii-Levenštein bound takes over. In this paper we improve Rogers' upper bound for the density of unit ball packings in Euclidean d-space for all d ≥ 8 and improve the Kabatjanskii-Levenštein upper bound in small dimensions. Namely, we show that the volume of any Voronoi cell in a packing of unit balls in E d , d ≥ 8, is at least ω d /σ d and so the density of any unit ball packing in E d , d ≥ 8, is at mostσ d , whereσ d is a geometrically well-defined quantity satisfying the inequalityσ d < σ d for all d ≥ 8. We prove this by showing that the surface area of any Voronoi cell in a packing of unit balls in E d , d ≥ 8, is at least (d · ω d )/σ d .