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Multidimensional companding quantization is analyzed theoretically for the case of high-resolution and mean-squared-error distortion. The optimality of choosing the expander function to be the inverse of the compressor function is established first. Heuristic derivations of the point density and moment of inertia of companding are given, which combined result in the distortion formula by Bucklew. Further, the interaction between the lattice quantizer (LQ) and compressor function is studied. Optimality is achieved with a second moment optimal LQ, shaped by a compressor-dependent linear transform. High rate theory for a radial compander and spherically symmetric sources is reviewed. The result is used to evaluate the performance on an independent and identically distributed (i.i.d.) Gaussian source. The radial compander clearly outperforms both a product scalar quantizer and a spherical quantizer for dimensions higher than two. Radial companding is also generalized to the correlated Gaussian source. Finally, a comparison of theory and practical companders is made in a Gaussian framework.