An arithmetic proof of John's ellipsoid theorem
  Using an idea of Voronoi in the geometric theory of positive definite quadratic forms, we give a transparent proof of John's characterization of the unique ellipsoid of maximum volume contained in a convex body. The same idea applies to the 'hard part' of a generalization of John's theorem and shows the difficulties of the corresponding 'easy part'. 
Peter M. Gruber, Franz E. Schuster
math.MG math.DG math.FA