Voronoi regions of lattices, second moments of polytopes, and quantization

Ah&act-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? In n-dimensional space, what is the average squared distance of a random point from the closest point of the lattice A, (or D,, , En, A: or D,*)? 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 Es has a mean-squared error per symbol of 0.0717 when applied to uniformly distributed data, compared with 0.0&333 . . for the best one:dimensional quantizer.