The Voronoi diagram for a set of K points in the plane is a partition of all the points of the plane into K regions, such that every point (of the plane) is assigned to the closest point among the K specified points. (See Figure 8.5.) What is the relationship between Voronoi diagrams and K-means clusters? What do Voronoi diagrams tell us about the possible shapes of K-means clusters?
(a) If we have K K-means clusters, then the plane is divided into K Voronoi
regions that represent the points closest to each centroid.
(b) The boundaries between clusters are piecewise linear. It is possible to
see this by drawing a line connecting two centroids and then drawing
a perpendicular to the line halfway between the centroids. This per-
pendicular line splits the plane into two regions, each containing points
that are closest to the centroid the region contains.
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