Lattice Points inside Lattice Polytopes
  We show that, if the interior of a lattice d-polytope P contains at least one lattice point, then it contains a lattice point whose coefficient of asymmetry with respect to P is at most b for some number b depending on d only.   As an application, we obtain new upper bounds on the volume of a lattice polytope given the number of lattice points in its interior. 
Oleg Pikhurko