# Equality of Voronoi diagram

What can we say about two sets $A$ and $B$ if both of them have the same Voronoi diagram.

First, I thought if the Voronoi diagram are equal so the sets also should be equal, but by definition, Voronoi diagram is determined by distances to a specified family of objects (subsets) in the space, so do the same distances mean the same sets?

Is $A = B$?

Or $\left | A \right | = \left | B \right |$?

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On the other hand, any Voronoi diagram with at least one vertex is generated by a unique set of sites. So if there is at least one Voronoi vertex, you really do have $A=B$. –  JeffE Jun 25 '12 at 13:28
@JeffE, equality of Voronoi edges and Voronoi vertices denotes the equality of sets $A=B$ –  fog Jul 1 '12 at 12:53