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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 greedyquantization 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 quantizationsequences of radial density distributions and show that they are $L^s$-rateoptimal for $s \in (r,r+d)$ under some moment assumption. Based on the resultsestablished in $\cite{Sagna08}$ for $L^r$-optimal quantizers, we show, for alarger 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 $s < r+d$. Weshow, 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 thisapproach to numerical integration.

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