On the classification of convex lattice polytopes
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Heling Liu
Abstract
In 1980, Arnold studied the classification problem for convex lattice polygons of given area. Since then this problem and its high dimensional analogue have been studied by Bárány, Pach, Vershik and others. Bounds for the number of non-equivalent d-dimensional convex lattice polytopes of given volume have been achieved. In this paper we study Arnold's problem for centrally symmetric lattice polygons and the classification problem for convex lattice polytopes of given cardinality. In the plane we obtain analogues to the bounds of Arnold, Bárány and Pach in both cases. However, the number of non-equivalent d-dimensional convex lattice polytopes of w lattice points is infinite whenever w – 1 ≥ d ≥ 3, which may intuitively contradict to Bárány and Vershik's upper bound.
© de Gruyter 2011
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- 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
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
