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 (see [14] and [15]). Finally semi-closed forms of $L^r$-optimal dual quantizers are established for power distributions on compacts intervals and truncated exponential distributions.

对于双重量化，我们建立了Kieffer唯一性结果的对等形式，用于紧密支持的一维概率分布，其中 $日志$-凹面密度（也称为强单峰）：对于此类分布， ${\mathrm{大号}}^{\mathrm{[R}}$-最佳双重量化器在每个级别都是唯一的 $ñ$，最佳网格是量化误差的唯一临界点。展示了一个临界点的唯一性失败的非强烈单峰分布的例子。在二次方$\mathrm{[R}=\mathrm{2个}$在这种情况下，我们提出了一种算法，用于计算唯一的最佳对偶量化器。它在Voronoi框架中提供了Lloyd方法I算法的对应物（请参见[14]和[15]）。最后的半封闭形式${\mathrm{大号}}^{\mathrm{[R}}$建立最佳双量化器，用于压缩间隔上的功率分布和截断的指数分布。