Conditional constrained and unconstrained quantization for probability distributions

In this paper, we introduce and develop the concept of conditional quantization for Borel probability measures on $\mathbb{R}^k,$ considering both constrained and unconstrained frameworks. For each setting, we define the associated quantization errors, dimensions, and coefficients, and provide explicit computations for specific classes of probability distributions. A key result in the unconstrained case is that the union of all optimal sets of $n$-means is dense in the support of the measure. Furthermore, we demonstrate that in conditional constrained quantization, if the conditional set is contained within the union of the constraint family, then the lower and upper quantization dimensions, as well as the corresponding coefficients, remain unaffected by the conditional set for any Borel probability measure. In contrast, if the conditional set is not contained within this union, these properties may no longer hold, as illustrated through various examples.