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

Optimal dual quantizers of $1D$ $\log$-concave distributions: uniqueness and Lloyd like algorithm