The inverse of a Voronoi diagram is a Delaunay triangulation. Instead of determining the spaces closest to the point, it is the lines that connect the points while avoiding skinny triangles. (The technical definition is that a Delaunay triangulation for a set P of points in a plane is a triangulation DT(P) such that no point in P is inside the circumcircle of any triangle in DT(P).)

http://en.wikipedia.org/wiki/Delaunay_triangulation

If you use a Voronoi diagram to generate a map of provinces with capital cities, then the Delaunay triangulation will tell you where to put the roads to connect the towns. It’s also useful if you have to subdivide a plane into something other than a grid and you want to keep the triangles spread out.

(image from http://en.wikipedia.org/wiki/File:Delaunay_circumcircles_vectorial.svg)