The local quantization behavior of absolutely continuous probabilities

For a large class of absolutely continuous probabilities P it is shown that, for r>0, for n-optimal Lr(P)-codebooks αn, and any Voronoi partition Vn,a with respect to αn the local probabilities P(Vn,a) satisfy P(Va,n)≈n−1 while the local Lr-quantization errors satisfy ∫Vn,a∥x−a∥rdP(x)≈n−(1+r/d) as long as the partition sets Vn,a intersect a fixed compact set K in the interior of the support of P.