Local Covering Optimality of the Leech Lattice
نویسندگان
چکیده
We show the highly non–surprising fact that the Leech lattice gives a sphere covering which is locally least dense among lattice coverings. This gives a first example of a locally optimal lattice covering having a non–simplicial Delone subdivision. Hereby, we in particular answer a question of Dickson posed in 1968. By showing that the Leech lattice is rigid, our answer is even strongest possible in a sense.
منابع مشابه
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We show that the Leech lattice gives a sphere covering which is locally least dense among lattice coverings. We show that a similar result is false for the root lattice E8. For this we construct a less dense covering lattice whose Delone subdivision has a common refinement with the Delone subdivision of E8. The new lattice yields a sphere covering which is more than 12% less dense than the form...
متن کامل. M G ] 1 5 O ct 2 00 4 Local Covering Optimality of Lattices : Leech Lattice versus Root Lattice
We show that the Leech lattice gives a sphere covering which is locally least dense among lattice coverings. We show that a similar result is false for the root lattice E8. For this we construct a less dense covering lattice whose Delone subdivision has a common refinement with the Delone subdivision of E8. The new lattice yields a sphere covering which is more than 12% less dense than the form...
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The Leech lattice has many strange properties, discovered by Conway, Parker, and Sloane. For example, it has covering radius √ 2, and the orbits of points at distance at least √ 2 from all lattice points correspond to the Niemeier lattices other than the Leech lattice. (See Conway and Sloane [6, Chaps. 22-28].) Most of the properties of the Leech lattice follow from the fact that it is the Dynk...
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