  We establish for dual quantization the counterpart of Kieffer's uniqueness
result for compactly supported one dimensional probability distributions having
a $\log$-concave density (also called strongly unimodal): for such
distributions, $L^r$-optimal dual quantizers are unique at each level $N$, the
optimal grid being the unique critical point of the quantization error. An
example of non-strongly unimodal distribution for which uniqueness of critical
points fails is exhibited.
  In the quadratic $r=2$ case, we propose an algorithm to compute the unique
optimal dual quantizer. It provides a counterpart of Lloyd's method~I algorithm
in a Voronoi framework. Finally semi-closed forms of $L^r$-optimal dual
quantizers are established for power distributions on compacts intervals and
truncated exponential distributions.
