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Lattice points in lattice polytopes

  • Oleg Pikhurko (a1)
  • DOI:
  • Published online: 01 February 2010

It is shown that, for any lattice polytope P⊂ℝd the set int (P)∩lℤd (provided that it is non-empty) contains a point whose coefficient of asymmetry with respect to P is at most 8d · (8l+7)22d+1. If, moreover, P is a simplex, then this bound can be improved to 8 · (8l+7 )2d+1. As an application, new upper bounds on the volume of a lattice polytope are deduced, given its dimension and the number of sublattice points in its interior.

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1.H. Blichfeldt . A new principle in the geometry of numbers with some applications. Trans. Amer. Math. Soc, 15 (1914), 227235.

2.D. Hensley . Lattice vertex polytopes with interior lattice points. Pacific. J. Math, 105 (1983). 183191.

3.J. C. Lagarias and G. M. Ziegler . Bounds for lattice polytopes containing a fixed number of interior points in a sublattice. Canad. J. Math., 43 (1991), 10221035.

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  • ISSN: 0025-5793
  • EISSN: 2041-7942
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