Use 1 or 2 norm for Voronoi vector quantization?
In general the quality of the quantization is measured using the mean squared error (MSE) between the input vector $x$ and the reproduction value $q(x)$ ($w_k$ in the notation you used in the question). For the MSE the best partition is the one defined by Voronoi regions under the euclidean distance ($l^2$ norm), i.e: 
$$V_j = \{ x \in R^d : \|x - w_j \|^2 \leq \|x - w_i \|^2\ for\ all\ i   \}$$

You can find the proof in here. Also here and here the euclidean distance is used.