Sufficient conditions for uniqueness of a locally optimal quantizer for a class of convex error weighting functions

Sufficient conditions are presented for uniqueness of a locally optimal quantizer being a stationary point of the quantization distortion measure , the expected value of an error weighting function , where is a random variable to be quantized, where the probability density function describing is continuous and positive on some finite or infinite interval and zero outside it, and where is the quantization error. The function is assumed convex and symmetric in and zero only for . It is shown that in the cases of and , the simple condition of concavity of In is sufficient for uniqueness of a locally optimal quantizer.