Lattice Points inside Lattice Polytopes
نویسنده
چکیده
We show that, for any lattice polytope P ⊂ R, the set int(P ) ∩ lZ (provided it is non-empty) contains a point whose coefficient of asymmetry with respect to P is at most 8d · (8l+7) 2d+1 . If, moreover, P is a simplex, then this bound can be improved to 9 · (8l+ 7) d+1 . This implies that the maximum volume of a lattice polytope P ⊂ R d containing exactly k ≥ 1 points of lZ in its interior, is p(d, k, l) ≤ (8dl) · (8l+7) 2d+1 · k. For the corresponding maximum s(d, k, l) for simplices, we obtain s(d, k, l) ≤ 9l · (8l + 7) d+1 · k/d!.
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