Abstract
The existence of optimal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> -level quantizers is demonstrated for several classes of quantizers including block quantizers, memoryless quantizers, and uniform step-size quantizers. No assumptions whatsoever are made regarding the input probability distribution. The distortion measure considered generalizes many of the commonly used error-based distortion measures, including mean squared-error distortion and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</tex> th law distortion.
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