Obtain the set of points from Voronoi diagram

Given a planar infinite two dimensional mesh graph such that each small polygon of the mesh is convex, is it correct to assume for any such mesh there exists a set of points such that the these polygons will be their corresponding Voronoi diagram? If so, how we can calculate it

Hint: It's false. Counter-example: $\boxed{\strut}\boxed{\strut\quad}\!\boxed{\strut}$. (Adjacent points must be equidistant from the edge.)