Packing and covering by translates of certain nonconvex bodies

  title={Packing and covering by translates of certain nonconvex bodies},
  author={Hugh Everett and Dean R. Hickerson},
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… 
A Reduction of Lattice Tiling by Translates of a Cubical Cluster
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
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
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
  • S. Stein
  • Mathematics, Computer Science
    IEEE 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.
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
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
  • S. Szabó
  • Mathematics, Computer Science
    Discret. 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
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
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
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


A symmetric star body that tiles but not as a lattice
A classical question in convex bodies runs as follows : "If translates of a fixed convex body K in Euclidean space can be packed with a certain density, is it possible to find a lattice packing by
Factoring groups and tiling space
The problem of tiling space by translates of certain star bodies, called ‘crosses’ and ‘semicrosses’, is intimately connected with finding a subsetA of a finite abelian groupG such that for a
Splitting groups by integers
A question concerning tiling Euclidean space by crosses raised this algebraic question: Let G be a finite abelian group and S a set of integers. When do there exist elements g1, g2 ... gn in G such
Tiling space by congruent polyhedra
The purpose of this department is to provide early announcement of outstanding new results, with some indication of proof. Research announcements are limited to 100 typed lines of 65 spaces each. A
Algebraic tiling
  • Amer. Math. Monthly
  • 1974
Factoring groups and tiling space, Aequationes Math
  • Factoring groups and tiling space, Aequationes Math
  • 1973
Factoring by subsets, Pacific
  • J. Math
  • 1967
Factoring by subsets
Packing and covering, Cambridge Tracts in Math, and Math
  • Physics
  • 1964
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