k-means clustering and Voronoi sets

In the context of https://latex.codecogs.com/gif.latex?k-means, we want to partition the space of our observations into https://latex.codecogs.com/gif.latex?k classes. each observation belongs to the cluster with the nearest mean. Here “nearest” is in the sense of some norm, usually the https://latex.codecogs.com/gif.latex?\ell_2 (Euclidean) norm.

Consider the case where we have 2 classes. The means being respectively the 2 black dots. If we partition based on the nearest mean, with the https://latex.codecogs.com/gif.latex?\ell_2 (Euclidean) norm we get the graph on the left, and with the https://latex.codecogs.com/gif.latex?\ell_1 (Manhattan) norm, the one on the right,

Points in the red region are closer to the mean in the upper part, while points in the blue region are closer to the mean in the lower part. Here, we will always use the standard https://latex.codecogs.com/gif.latex?\ell_2 (Euclidean) norm. Note that the graph above is related to Voronoi diagrams (or Voronoy, from Вороний in Ukrainian, or Вороно́й in Russian) with 2 points, the 2 means.

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