Uniqueness of locally optimal quantizer for log-concave density and convex error weighting function

It is desired to encode a random variable using an -level quantizer to minimize the expected distortion , where the error weighting function is convex, strictly increasing and continuously differentiable. It is shown that if has a log-concave density, then there exists a unique locally optimal quantizer and Lloyd\´s Method I may be used to find . Trushkin had earlier shown this result for the error weighting functions and .