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

If a point is picked at random inside a regular simplex, octahedron, -cell, or other polytope, what is its average squared distance from the centroid? In -dimensional space, what is the average squared distance of a random point from the closest point of the lattice (or 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 of when applied to uniformly distributed data, compared with for the best one-dimensional quantizer.