Kantorovich duality

Kantorovich duality is the fundamental theorem of optimal transport theory, established by Leonid Kantorovich in 1942, which reformulates the problem of finding the most efficient way to transport mass from one distribution to another as a dual optimization problem over pairs of potential functions. The primal problem — minimizing the total cost of transport over all possible couplings of two probability measures — is infinite-dimensional and computationally intractable. The dual problem — maximizing the difference of integrated potentials subject to a constraint on their difference — is often dramatically simpler and reveals structural properties invisible in the primal formulation.

The duality states that under mild regularity conditions, the minimal transport cost equals the maximal dual value. This equality is not merely a computational convenience; it is a deep structural result that connects optimal transport to the theory of linear programming, convex analysis, and partial differential equations. The optimal potentials in the dual problem encode the geometric structure of the optimal transport map: where the potentials are differentiable, the transport map is given by the gradient of a convex function, a result known as Brenier's theorem.

Kantorovich duality has become one of the most powerful tools in modern analysis, with applications ranging from economics (matching problems, mechanism design) to machine learning (Wasserstein generative adversarial networks) to geometry (synthetic notions of Ricci curvature via Lott-Sturm-Villani theory). The duality reveals that optimal transport is not merely a problem in logistics but a lens through which the geometry of probability spaces becomes visible.