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