Is a Voronoi diagram a partitioning of the plane?

Question

The Wikipedia article on Voronoi diagrams suggests that it is a partitioning of a plane. But locations on the plane that are equidistant from two generator points, the cell boundaries, are surely assigned to both cells or neither, suggesting to me that it doesn't meet the formal criteria of a partition that every point belong to exactly one cell. Is this why it's called a partitioning but not a partition? Are the two distinctly defined?

Answer 1

Usually a partition of a set $A$ is a collection $(A_i)_{i\in I}$ of disjoint nonempty subsets $A_i\subset A$ such that $\bigcup_{i\in I}A_i=A$. Now the Voronoi cells $V_i$ $(1\leq i\leq n)$ created by a finite point set $\{x_1,x_2,\ldots, x_n\}\subset {\mathbb R}^2$ do not form a partition of ${\mathbb R}^2$ in this strict sense, insofar as it would be very difficult to attribute the common boundary points of several $V_i$s to the individual $V_i$s in a coherent way. By abuse of language one nevertheless says that the $V_i$ form a partition of ${\mathbb R}^2$, because everyone involved knows very well what is happening here. Talking about "partitionings" vs. "partitions" is just a smoke screen obscuring the matter.