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

A lattice polytope is a polytope in R sup n whose vertices are all in Z sup n. The volume of a lattice polytope P containing exactly k >= 1 points in dZ sup n in its interior is bounded above by kd sup n (7(kd + 1))n sup 2 sup (n+1). Any lattice polytope in R sup n 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 polytopes of bounded volume is finite.