Rényi entropy and quantization for densities

A random variable Z taking value in a finite, nonatomic measure space (X;M; μ) and whose distribution is absolutely continuous with respect to μ is to be described using N labels. We seek the labeling that minimizes the ρ-th moment of the μ-volume of the set of points in X that have the same label as Z. The large-N asymptotics of this minimum are expressed in terms of the Rényi entropy of order 1=(1 + ρ).