Given sites in a Euclidean space, the Voronoi cell of a site is the set of points at least as close to it as to every other site. The Voronoi diagram is the subdivision formed by all of these cells.
For centres in Euclidean space, the Voronoi cell of is the set of points at least as close to as to every other centre. Shared boundaries consist of points tied between nearest centres; assigning each such point to one centre makes the cells a partition of a set.
On the real line with distinct ordered centres, the Voronoi cells are intervals, with boundaries at the midpoints of adjacent centres. This makes nearest-centre clusters contiguous blocks of sorted observations in k-means clustering.
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A Voronoi diagram is a mathematical structure that partitions a space into regions based on the distance to a specific set of points, called seed points or sites. Each region in a Voronoi diagram corresponds to one of the seed points, and every point within that region is closer to its associated seed point than to any other seed point.