Optimally solving a transportation problem using Voronoi diagrams

We consider the following variant of the well-known Monge-Kantorovich transportation problem. Let S be a set of n point sites in Rd. A bounded set C⊂Rd is to be distributed among the sites p∈S such that (i) each p receives a subset Cp of prescribed volume and (ii) the average distance of all points z of C from their respective sites p is minimized. In our model, volume is quantified by a measure μ, and the distance between a site p and a point z is given by a function dp(z). Under quite liberal technical assumptions on C and on the functions dp(⋅) we show that a solution of minimum total cost can be obtained by intersecting with C the Voronoi diagram of the sites in S, based on the functions dp(⋅) equipped with suitable additive weights. Moreover, this optimum partition is unique, up to sets of measure zero. Unlike the deep analytic methods of classical transportation theory, our proof is based directly on simple geometric arguments.