What is the intuition behind the Kantorovich potential in optimal transport?

Question

From what I currently understand, under certain conditions one may turn the usual Kantorovich problem - a minimisation problem in terms of measures into a maximisation problem in terms of functions. By “turn into” I mean that the optimal values for both problems agree.

The Kantorovich potential associated to the problem is the function $\phi$ that achieves the maximum in the latter problem, and can be chosen to be $c$-concave where $c$ is the cost function. To be more precise, the latter problem is a minimisation over pairs of functions, and the solution can be taken to be $(\phi, \phi^c)$ for a $c$-concave function $\phi$, where the superscript $c$ denotes the $c$-transform.

Many results about the original problem can be proven by examining the Kantorovich potential. While I understand the proofs formally, I cannot visualise what exactly a Kantorovich potential is doing. Is there a geometric/analytic interpretation of the potential? For example, how does it relate directly to the optimal transport plan? Can one deduce the geometry of the plan from the potential and vice versa?

Answer 1

I recommend the interpretation with bakeries and cafes! In Villani's "Optimal Transport Old and New" in Chapter 5 "Cyclic monotonicity and Kantorovich duality you'll find this:

I shall start by explaining the concepts of cyclical monotonicity and Kantorovich duality in an informal way, sticking to the bakery analogy of Chapter 3. Assume you have been hired by a large consortium of bakeries and cafés, to be in charge of the distribution of bread from production units (bakeries) to consumption units (cafés). The locations of the bakeries and cafés, their respective production and consumption rates, are all determined in advance. You have written a transference plan, which says, for each bakery (located at) $x_i$ and each café $y_j$ , how much bread should go each morning from $x_i$ to $y_j$. [...]

The next key concept is the dual Kantorovich problem. While the central notion in the original Monge–Kantorovich problem is cost, in the dual problem it is price. Imagine that a company offers to take care of all your transportation problem, buying bread at the bakeries and selling them to the cafés; what happens in between is not your problem (and maybe they have tricks to do the transport at a lower price than you). Let $ψ(x)$ be the price at which a basket of bread is bought at bakery $x$, and $φ(y)$ the price at which it is sold at café $y$. On the whole, the price which the consortium bakery+café pays for the transport is $φ(y) − ψ(x)$, instead of the original cost $c(x, y)$. This of course is for each unit of bread: if there is a mass $μ(dx)$ at $x$, then the total price of the bread shipment from there will be $ψ(x) μ(dx)$. So as to be competitive, the company needs to set up prices in such a way that $$∀(x, y),\ φ(y) − ψ(x) ≤ c(x, y).$$ When you were handling the transportation yourself, your problem was to minimize the cost. Now that the company takes up the transportation charge, their problem is to maximize the profits. This naturally leads to the dual Kantorovich problem: $$\sup\{\int_Y \varphi(y) d\nu(y) - \int_X \psi(x) d\mu(x):\ φ(y) − ψ(x) ≤ c(x, y)\}$$.