For topological combinatorics, Voronoi diagrams provide extremely nice configuration spaces. In particular, some mass partitioning problems can be tackled using this type of subdivision. Power diagrams (which extend Voronoi diagrams) are more commonly used for this purpose. See for instance the expository article of Günter Ziegler  where he explains his results from . The problem of interest in those papers is the following conjecture of R. Ramana Rao
"Given any convex shape in the plane and any positive integer N. There exist some way(s) of partitioning this shape into N convex pieces so that they all have the same area and perimeter."
 Günter M. Ziegler. "Cannons at sparrows" Newsletter of the European Mathematical Society 95: 25-31
 Blagojević, Pavle VM, and Günter M. Ziegler. "Convex equipartitions via equivariant obstruction theory." Israel Journal of Mathematics 200.1 (2014): 49-77.