  We provide a complete picture of the upper quantization dimension in terms of
the R\'enyi dimension by proving that the upper quantization dimension
$\bar{D}_{r}(\nu)$ of order $r>0$ for an arbitrary compactly supported Borel
probability measure $\nu$ is given by its R\'enyi dimension at the point
$q_{r}$ where the $L^{q}$-spectrum of $\nu$ and the line through the origin
with slope $r$ intersect. In particular, this proves the continuity of
$r\mapsto\bar{D}_{r}(\nu)$ as conjectured by Lindsay (2001). This viewpoint
also sheds new light on the connection of the quantization problem with other
concepts from fractal geometry in that we obtain a one-to-one correspondence of
the upper quantization dimension and the $L^{q}$-spectrum restricted to
$\left(0,1\right)$. We give sufficient conditions in terms of the
$L^{q}$-spectrum for the existence of the quantization dimension. In this way
we show as a byproduct that the quantization dimension exists for every Gibbs
measure with respect to a $\mathcal{C}^{1}$-self- conformal iterated function
system on $\mathbb{R}^{d}$ without any assumption on the separation conditions
as well as for inhomogeneous self-similar measures under the inhomogeneous open
sets condition. Some known general bounds on the quantization dimension in
terms of other fractal dimensions can readily be derived from our new approach,
some can be improved.
