Voronoi regions of lattices, second moments of polytopes, and quantization
J. Conway, Neil J.A. Sloane · IEEE Transactions on Information Theory · 1982
If a point is picked at random inside a regular simplex, octahedron,600-cell, or other polytope, what is its average squared distance from the centroid? Inn-dimensional space, what is the average squared distance of a random point from the closest point of the latticeA_{n}(orD_{n}, E_{n}, A_{n}^{\ast} or D_{n}^{\ast})?The answers are given here, together with a description of the Voronoi (or nearest neighbor) regions of these lattices. The results have applications to quantization and to the design of signals for the Gaussian channel. For example, a quantizer based on the eight-dimensional lattice E8 has a mean-squared error per symbol of0.0717 \cdotswhen applied to uniformly distributed data, compared with0.08333 \cdotsfor the best one-dimensional quantizer.