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Uniqueness of lattice packings and coverings of extreme density
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Peter M. Gruber
Published/Copyright:
August 12, 2011
Abstract
For d = 2, 3 a generic convex body in 
has a unique lattice packing and for d ≥ 4 at most a(d) lattice packings of maximum density, where a(d) ≥ 1 is a constant. If in a certain connectedness property holds, one may take a(d) = 1. Dually, for d = 2 a generic convex body has a unique lattice covering of minimum density and for d ≥ 3 there is a constant b(d) ≥ 1 such that it has at most b(d) such coverings.
Key words.: Convex body; lattice packing; lattice covering; extreme density; extreme lattice; uniqueness; Baire categories
Received: 2010-10-29
Published Online: 2011-08-12
Published in Print: 2011-November
© de Gruyter 2011
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Keywords for this article
Convex body;
lattice packing;
lattice covering;
extreme density;
extreme lattice;
uniqueness;
Baire categories
Articles in the same Issue
- Quasigeodesics and farthest points on convex surfaces
- The geometry of canal surfaces and the length of curves in de Sitter space
- On the mutual position of two irreducible conics in PG(2, q), q odd
- On the symmetric average of a convex body
- Ample vector bundles and polarized manifolds of sectional genus three
- Flat Laguerre planes of Kleinewillinghöfer type III.B
- Quotients of hypersurfaces in weighted projective space
- Characterizing the mixed volume
- Uniqueness of lattice packings and coverings of extreme density
- On the classification of convex lattice polytopes
- Blichfeldt-type inequalities and central symmetry
- Valuations on function spaces
