$L^s$-rate optimality of dilated$/$contracted $L^r$-optimal and greedy quantization sequences

We investigate some $L^s$-rate optimality properties of dilated/contracted $L^r$-optimal quantizers and $L^r$-greedy quantization sequences $(\alpha^n)_{n \geq 1}$ of a random variable $X$. We establish, for different values of $s$, $L^s$-rate optimality results for $L^r$-optimally dilated/contracted greedy quantization sequences $(\alpha^n_{\theta,\mu})_{n \geq 1}$ defined by $\alpha^n_{\theta,\mu}=\{\mu+\theta (\alpha_i-\mu), \alpha_i \in \alpha^{(n)}\}$. We lead a specific study for $L^r$-optimal greedy quantization sequences of radial density distributions and show that they are $L^s$-rate optimal for $s \in (r,r+d)$ under some moment assumption. Based on the results established in $\cite{Sagna08}$ for $L^r$-optimal quantizers, we show, for a larger class of distributions, that the dilatation $(\alpha^n_{\theta,\mu})_{n \geq 1}$ of an $L^r$-optimal quantizer is $L^s$-rate optimal for $s < r+d$. We show, for various probability distributions, that there exists a parameter $\theta^*$ for which the dilated quantization sequence satisfy the so-called {\em $L^s$-empirical measure} theorem and present an application of this approach to numerical integration.