Asymptotic performance of block quantizers with difference distortion measures

Gersho's bounds on the asymptotic (large rate or small distortion) performance of block quantizers are valid for vector distortion measures that are powers of the Euclidean or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l₂</tex> norm. These results are generalized to difference distortion measures that are increasing functions of the seminorm of their argument, where any seminorm is allowed. This provides a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -dimensional generalization of Gish and Pierce's results for single-symbol quantizers. When the distortion measore is a power of a seminorm the bounds are shown to be strictly better than the corresponding bounds provided by the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> th-order rate-distortion functions.