  For a large class of absolutely continuous probabilities $P$ it is shown
that, for $r>0$, for $n$-optimal $L^r(P)$-codebooks $\alpha_n$, and any Voronoi
partition $V_{n,a}$ with respect to $\alpha_n$ the local probabilities
$P(V_{n,a})$ satisfy $P(V_{a,n})\approx n^{-1}$ while the local
$L^r$-quantization errors satisfy $\int_{V_{n,a}}|x-a|^r dP(x)\approx
n^{-(1+r/d)}$ as long as the partition sets $V_{n,a}$ intersect a fixed compact
set $K$ in the interior of the support of $P$.
