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

If a point is picked at random inside a regular simplex, octahedron, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">600</tex> -cell, or other polytope, what is its average squared distance from the centroid? In <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -dimensional space, what is the average squared distance of a random point from the closest point of the lattice <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Aₙ</tex> (or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Dₙ, Eₙ, Aₙ^{} or Dₙ^{})?</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0.0717 ⋯</tex> when applied to uniformly distributed data, compared with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0.08333 ⋯</tex> for the best one-dimensional quantizer.