# Packing and covering by translates of certain nonconvex bodies

@inproceedings{Everett1979PackingAC, title={Packing and covering by translates of certain nonconvex bodies}, author={Hugh Everett and Dean R. Hickerson}, year={1979} }

We develop techniques for determining the packing and covering constants for star bodies composed of cubes. In the theory of convex sets problems of tiling, packing, and covering by translates of a given set have a long history, with the main focus on the packing and covering by spheres. Only in a few cases is the densest packing or sparsest covering known, even in the case of the sphere, except, of course, when the set happens to tile Eucidean space. In a series of papers S. K. Stein [4], [5…

## 19 Citations

A Reduction of Lattice Tiling by Translates of a Cubical Cluster

- 2005

A cluster is the union of a finite number of cubes from the standard partition of n-dimensional Euclidean space into unit cubes. If there is lattice tiling by translates of a cluster, then must there…

Combinatorial packings of R3 by certain error spheres

- Mathematics, Computer ScienceIEEE Trans. Inf. Theory
- 1984

One of the "error spheres" discussed by Golomb in 1969, his "Stein corner" in three-dimensional Euclidean space R3 is concerned, and sufficiently dense packings are produced to show that they are much denser than the densest lattice packing.

On covering by translates of a set

- Mathematics, Computer ScienceRandom Struct. Algorithms
- 2011

It is shown that if n(k) grows at a suitable rate with k, then almost every k-subset of any given group with order n comes close to the optimal efficiency of this minimal number τ(S,G) of an arbitrary subset S of a group G.

Packings of Rn by certain error spheres

- Mathematics, Computer ScienceIEEE Trans. Inf. Theory
- 1984

It is shown that packings by the cross that are extremely regular (lattice packings) do just about as well as arbitrary packings in all dimensions and for k large, and for the semicross, even in R^{3} , when the arm length k is large, latticePackings are much less dense than arbitraryPackings.

TILING 3 AND 4-DIMENSIONAL EUCLIDEAN SPACES BY LEE SPHERES

- 2017

The paper addresses the problem if the n-dimensional Euclidean space can be tiled with translated copies of Lee spheres of not necessarily equal radii such that at least one of the Lee spheres has…

Rational tilings by -dimensional crosses

- Mathematics
- 1983

Consider the set of closed unit cubes whose edges are parallel to the coordinate unit vectors e,,...,e" and whose centers are re., 0 k for every prime divisor p of 2 kn + I, then there is a rational…

A reduction of lattice tiling by translates of a cubical cluster

- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 1987

If the interior of the cluster is connected and the dimension is at most three, then the answer is affirmative and there is lattice tiling by translates of a cluster in which the translation vectors have only integer coordinates.

Tilings by (0.5, n)-Crosses and Perfect Codes

- Mathematics, Computer ScienceSIAM J. Discret. Math.
- 2013

It is proved that an integer tiling for such a shape exists if and only if $ n=2^t-1$ or $n=3^t -1$, where $t>0$.

Tilings with Generalized Lee Spheres

- Mathematics
- 2003

We discuss tilings of ℝ n with bodies which are generalizations of the well known Lee spheres. It is shown that if n=2 then there exists a tiling of ℝ n with any generalized Lee spheres of order n.…

Covering abelian groups with cyclic subsets

- Mathematics
- 1995

SummaryLetk andm be positive integers. An abelian groupG is said to have ann-cover if there is a subsetS ofG consisting ofn elements such that every non-zero element ofG can be expressed in the…

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