On quantization with the Weaire-Phelan partition

Until recently, the solution to the Kelvin problem of nding a partition of R 3 into equal-volume cells with the least surface area was believed to be tessellation by the truncated octahe-dron. In 1994, D. Weaire and R. Phelan described a partition that outperformed the truncated octahedron partition in this respect. In this paper, we show that the eeective normalized moment of inertia (NMI) of the Weaire-Phelan (WP) partition is larger than that of the truncated octahedron partition, and so does not provide a counterexample to Gersho's conjecture that the truncated octahedron partition has the least eeective NMI among all partitions of R 3. We also show that if the WP partition is used as the partition of a 3-dimensional vector quantizer, with the corresponding codebook consisting of the centroids of the cells, then the resulting quantiza-tion error is white. As a result, the eeective NMI of the WP partition cannot be reduced by passing it through an invertible linear transformation.