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Bounds for lattice polytopes containing a fixed number of interior points in a sublattice. (English) Zbl 0752.52010
A lattice polytope is a convex polytope in \(\mathbb{R}^ n\) whose vertices are all in \(\mathbb{Z}^ n\). The authors show that the volume of a lattice polytope \(P\) with \(k=\text{card}(\mathbb{Z}^ d\cap\text{int }K)\geq 1\) is bounded above by \(k(7(k+1))^{n2^{n+1}}\), which improves an earlier result by D. Hensley (1983).
Zaks, Perles and the reviewer (1982) gave lower bounds, which show that the order of magnitude of the upper bound is essentially best possible.
Reviewer: J.M.Wills (Siegen)

52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
11H06 Lattices and convex bodies (number-theoretic aspects)
11P21 Lattice points in specified regions
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