On the Voronoi Regions of Certain Lattices

The Voronoi region of a lattice Lₙ ⊆ Rⁿ is the convex polytope consisting of all points of Rⁿ that are closer to the origin than to any other point of Lₙ. In this paper we calculate the second moments of the Voronoi regions of the lattices E₆^ * ,E₇^ * ,K₁₂ ,Λ ₁₆ and Λ ₂₄. The results show that these lattices are the best quantizers presently known in dimensions 6, 7,12,16 and 24. The calculations are performed by Monte Carlo integration, and make use of fast algorithms for finding the closest lattice point to an arbitrary point of the space. We also establish two general theorems concerning the number of faces of the Voronoi region of a lattice.