Worst-case rate of scalar vs. vector quantization

We show that there can be an arbitrary discrepancy between the worst-case rate required for scalar and vector quantization. Specifically, that for every δ, however large, and every ε>0, however small, there is a random variable and a distortion measure where quantization of a single instance within a given distortion requires more than δ bits in the worst case, but quantization of multiple independent instances within the same distortion requires at most ε bits per instance in the worst case. Furthermore, these discrepancies can be achieved by simple distortion measures that attain just two values: 0 and ∞