 Voronoi cell

For centres $m_1,\dots ,m_r$ in Euclidean space, the Voronoi cell of $m_j$ is the set of points at least as close to $m_j$ 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.