Densest lattice packings of 3-polytopes

@article{Betke2000DensestLP,
  title={Densest lattice packings of 3-polytopes},
  author={Ulrich Betke and Martin Henk},
  journal={Comput. Geom.},
  year={2000},
  volume={16},
  pages={157-186}
}
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70 Citations
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70 Citations

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References

SHOWING 1-10 OF 55 REFERENCES
An optimal convex hull algorithm in any fixed dimension
TLDR
A deterministic algorithm for computing the convex hull of n points inEd in optimalO(n logn+n⌞d/2⌟) time and a by-product of this result is an algorithm for Computing the Voronoi diagram ofn points ind-space in optimal O(nLogn+ n⌜d/ 2⌝) time.
On the densest packing of sections of a cube (1)
Sunto.Determinazione dei reticolati critici relativi alla porzione di un cubo compresa fra due piani perpendicolari ad una diagonale e simmetrici rispetto al centro.
THE DENSEST LATTICE PACKING OF TETRAHEDRA
The problem of finding the densest packing of tetrahedra was first suggested by Hubert [3, p. 319]. Minkowski [4] attempted to find the densest lattice packing of tetrahedra, but his result is…
Packing and Covering the Plane with Translates of a Convex Polygon
The Geometry of Numbers
THE meeting of the London Mathematical Society on December 19, 1946, took the form of a symposium on the geometry of numbers, arranged by Prof. H. Davenport. Prof. E. C. Titchmarsh, president of the…
Convex bodies which tile space by translation
It is shown that a convex body K tiles E d by translation if, and only if, K is a centrally symmetric d -polytope with centrally symmetric facets, such that every belt of K (consisting of those of…
Packing and Covering with Convex Sets
Lattice Octahedra
  • L. Mordell
  • Mathematics
    Canadian Journal of Mathematics
  • 1960
Let Ai, A2 , … , An be n linearly independent points in n-dimensional Euclidean space of a lattice Λ. The points ± A1, ±A2 , . . , ±An define a closed n-dimensional octahedron (or “cross poly tope“)…
Lectures on Polytopes
Based on a graduate course given at the Technische Universitat, Berlin, these lectures present a wealth of material on the modern theory of convex polytopes. The clear and straightforward…
On the Frustrum of a Sphere
(1) K: f(x,y,z) _ 1, which is symmetrical about 0, defines a convex body K say, in a three-dimensional Euclidean space. Let A denote any lattice of points admissible for K, so that all points of A,…
...
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