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Delaunay polytopes derived from the Leech lattice | | Mathieu Dutour Sikiric
; Konstantin Rybnikov
; | | Date: |
4 Jul 2009 | | Abstract: | Given a lattice L of R^n, a polytope D is called a Delaunay polytope in L if
the set of its vertices is Scap L where S is a sphere having no lattice points
in its interior. D is called perfect if the only ellipsoid in R^n that contains
Scap L is exactly S.
For a vector v of the Leech lattice Lambda_{24} we define Lambda_{24}(v) to
be the lattice of vectors of Lambda_{24} orthogonal to v. We studied Delaunay
polytopes of L=Lambda_{24}(v) for |v|^2<=22. We found some remarkable examples
of Delaunay polytopes in such lattices and disproved a number of long standing
conjectures. In particular, we discovered:
--Perfect Delaunay polytopes of lattice width 4; previously, the largest
known width was 2.
--Perfect Delaunay polytopes in L, which can be extended to perfect Delaunay
polytopes in superlattices of L of the same dimension.
--Polytopes that are perfect Delaunay with respect to two lattices $Lsubset
L’$ of the same dimension.
--Perfect Delaunay polytopes D for L with |Aut L|=6|Aut D|: all previously
known examples had |Aut L|=|Aut D| or |Aut L|=2|Aut D|.
--Antisymmetric perfect Delaunay polytopes in L, which cannot be extended to
perfect (n+1)-dimensional centrally symmetric Delaunay polytopes.
--Lattices, which have several orbits of non-isometric perfect Delaunay
polytopes.
Finally, we derived an upper bound for the covering radius of
Lambda_{24}(v)^{*}, which generalizes the Smith bound and we prove that it is
met only by Lambda_{23}^{*}, the best known lattice covering in R^{23}. | | Source: | arXiv, 0907.0776 | | Services: | Forum | Review | PDF | Favorites | |
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