Use 1 or 2 norm for Voronoi vector quantization? I have a script from a lecture. Basically it says that based on the Voronoi partitioning we identify the corresponding (nearest) class $w_k$ to a vector $x$ where $\left| {{w_k} - x} \right| = \mathop {\min }\limits_i \left( {\left| {{w_i} - x} \right|} \right)$ given the classes $w$.

The script uses the absolute value notation. This does not make much sense as we are using vectors. Which vector norms can or should be used? The 1 or 2 norm?
The 1 norm is faster to compute but maybe there are drawbacks I cannot currently think of.
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.