We provide a full picture of the upper quantization dimension in term of the
R\'enyi dimension, in that we prove that the upper quantization dimension of
order r>0 for an arbitrary compactly supported Borel probability measure
ν is given by its R\'enyi dimension at the point qr where the
Lq-spectrum of ν and the line through the origin with slope r
intersect. In particular, this proves the continuity of
r↦Dr 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 Lq-spectrum restricted to
(0,1). We give sufficient conditions in terms of the
Lq-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 C1-self-conformal iterated function
system on Rd 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.Comment: 17 pages, 1 figur