Z-1 High-Resolution Analysis of the Least Distortion of Fixed-Rate Vector Quantizers: Zador's Formula We wish to use high-resolution analysis to determine an approximate formula for the distortion of an optimal k-dimensional quantizer with dimension k and size M, assuming M is large. That is, we wish to find an approximate formula for the OPTA function δ(k,M) when M is large. To do this, we begin with the Bennett integral approximation to distortion: D ≅ 1 M 2/k ⌡ ⌠ m( x) λ 2/k ( x) f X ( x) d x A first thought is to find functions λ( x) and m( x) that minimize Bennett's integral, and then substitute these into the integral to find the least possible distortion. This is indeed a reasonable approach, which we shall take. However, we need to take into account the fact that λ( x) and m( x) are not arbitrary functions. Rather they represent a quantization density and an inertial profile, respectively. Thus, we must make sure to minimize Bennett's integral over the set of functions λ( x) and m( x) that are potential quantization densities and inertial profiles, respectively. On the one hand it is easy to say what functions are potential quantization densities is -- any nonnegative Z-2 function that integrates to one is a potential quantization density. On the other hand, it is more challenging to say what functions are potential inertial profiles. We would obviously like m( x) to be as small as possible. But it is not easy to say how small it can be. In what follows we first show that the best inertial profile is a constant. We next find the best quantization density. Finally we substitute these into Bennett's integral to determine what is called Zador's formula as an approximation to the OPTA function δ(k,M).