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Bounds for Lattice Polytopes Containing a Fixed Number of Interior Points in a Sublattice

  • Jeffrey C. Lagarias (a1) and Günter M. Ziegler (a2)
Abstract

A lattice polytope is a polytope in whose vertices are all in . The volume of a lattice polytope P containing exactly k ≥ 1 points in d in its interior is bounded above by . Any lattice polytope in of volume V can after an integral unimodular transformation be contained in a lattice cube having side length at most n ˙ n ! V. Thus the number of equivalence classes under integer unimodular transformations of lattice poly topes of bounded volume is finite. If S is any simplex of maximum volume inside a closed bounded convex body K in having nonempty interior, then K ( n + 2)S — (n+ l)s where mS denotes a nomothetic copy of S with scale factor m, and s is the centroid of S.

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References
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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