1 results
Bounds for Lattice Polytopes Containing a Fixed Number of Interior Points in a Sublattice
-
- Journal:
- Canadian Journal of Mathematics / Volume 43 / Issue 5 / 01 October 1991
- Published online by Cambridge University Press:
- 20 November 2018, pp. 1022-1035
- Print publication:
- 01 October 1991
-
- Article
-
- You have access
- Export citation
-
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.
whose vertices are all in 
. The volume of a lattice polytope 
. Any lattice polytope in 