Book ChapterDOI
Packing and Covering with Convex Sets
Gabor Toth,Wlodzimierz Kuperberg +1 more
- pp 799-860
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TLDR
The notions of packing and covering are well defined in hyperbolic geometry as mentioned in this paper, and packing in or covering of the whole space is called packing or a covering, respectively, and an arrangement that is a packing and a covering at the same time is called a tiling.Abstract:
Publisher Summary This chapter describes packing and covering with convex sets and discusses arrangements of sets in a space E, which should have a structure admitting the notions of congruence, measure, and convexity. Given a domain in E, a packing in the domain is an arrangement the members of which are all contained in the domain and have mutually disjoint interiors, and a covering of the domain is an arrangement whose union contains the domain. A packing in or a covering of the whole space E is called a packing or a covering, respectively. An arrangement that is a packing and a covering at the same time is called a tiling. All the known proofs of the theorem of Minkowski–Hlawka and its refinements are nonconstructive. The concepts of packing and covering are well defined in hyperbolic geometry. While high-density packings and low-density coverings can be considered efficient, the definition of density allows some undesired local deviations to occur that go contrary to the intuitive concept of efficiency.read more
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Journal ArticleDOI
Packing and Covering. By C. A. Rogers. Pp. viii, 111. 30s. 1964. (Cambridge)
TL;DR: The existence of reasonably dense lattice coverings and reasonably economical lattice covers has been studied in this paper, where the authors show that simplices cannot be very dense and coverings with spheres cannot have very economical coverings.
Journal ArticleDOI
Asymptotic estimates for best and stepwise approximation of convex bodies III
Stefan Glasauer,Peter M. Gruber +1 more
TL;DR: In this article, the authors considered approximations of a smooth convex body by inscribed and circumscribed convex polytopes as the number of vertices tends to infinity.
Journal ArticleDOI
Kepler’s conjecture: How some of the greatest minds in history helped solve one of the oldest math problems in the world
TL;DR: In this paper, it was shown that hexagonal close packing is the densest packing possible, and the density is now 74.05 percent, which is the highest known density.
Journal ArticleDOI
Densest lattice packings of 3-polytopes
Ulrich Betke,Martin Henk +1 more
TL;DR: In this article, the density of a densest lattice packing of an arbitrary 3-polytope is computed based on Minkowski's work on critical lattices of 3-dimensional convex bodies.
Posted Content
Densest Lattice Packings of 3-Polytopes
Ulrich Betke,Martin Henk +1 more
TL;DR: Based on Minkowski's work on critical lattices of 3-dimensional convex bodies, an efficient algorithm for computing the density of a densest lattice packing of an arbitrary 3-polytope is presented.
References
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Journal ArticleDOI
Packing and covering
TL;DR: The existence of reasonably dense lattice coverings and reasonably economical lattice covers has been studied in this article, where the authors show that simplices cannot be very dense and coverings with spheres cannot have very economical coverings.
Journal ArticleDOI
The Number of Circles Covering a Set
TL;DR: In this paper, it is shown that the best way to cover a given area with circles of a given radius E is to place the centers of the circles on an equilateral triangle network, i.e., to circumscribe (inscribe) the circles about the hexagons of a regular hexagon network or honeycomb.
Journal ArticleDOI
New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities
TL;DR: With the Delsarte-MacWilliams inequalities as a starting point, an upper bound is obtained on the rate of a binary code as a function of its minimum distance, which is asymptotically less than Levenshtein's bound and so also Elias's.