  Properties of scalar quantization with $r$th power distortion and constrained
R\'enyi entropy of order $\alpha\in (0,1)$ are investigated. For an
asymptotically (high-rate) optimal sequence of quantizers, the contribution to
the R\'enyi entropy due to source values in a fixed interval is identified in
terms of the "entropy density" of the quantizer sequence. This extends results
related to the well-known point density concept in optimal fixed-rate
quantization. A dual of the entropy density result quantifies the distortion
contribution of a given interval to the overall distortion. The distortion loss
resulting from a mismatch of source densities in the design of an
asymptotically optimal sequence of quantizers is also determined. This extends
Bucklew's fixed-rate ($\alpha=0$) and Gray \emph{et al.}'s variable-rate
($\alpha=1$) mismatch results to general values of the entropy order parameter
$\alpha$.
