20 1. The Kantorovich Duality Moreover, the infimum in the left-hand side of (1.4) is attained. Fur- thermore, it does not change the value of the supremum in the right-hand side of (1.4) if one restricts the definition of Φc to those functions (ϕ, ψ) which are bounded and continuous. We recall that a function F on a metric space X is said to be lower semi- continuous if it satisfies the characteristic property that, for any x X, F (x) lim inf y→x F (y). Let us make a few comments before starting the proof of Theorem 1.3. Further remarks will be made after the proof. Remarks 1.4. (i) It is not a priori clear that the value of sup J does not change upon restricting the definition of Φc to continuous functions, since it is not so clear that pairs of L1 functions satisfying (1.3) can be approximated by pairs of continuous functions satisfying the same inequality. However Remark 1.6 below will make this claim plausible. When we wish to emphasize the distinction between these definitions, we shall write informally Φc Cb, Φc L1. (ii) Actually, it is the infimum problem in (1.4) which should be called the dual problem (what will be used in the proof, is that M(X × Y ) is the dual space to C(X × Y ) when X, Y are compact!). (iii) Theorem 1.3 does not even assume that the value of the infimum is finite. (iv) As we shall see in the next section, in the case of a distance cost function, there is more to say about this duality. (v) In this chapter, we do not care whether the supremum of J is achieved or not. This question will be addressed in Chapter 2, see in particular Exercise 2.36. 1.1.3. The shipper’s problem. Here is an informal interpretation of The- orem 1.3, which we learnt from Caffarelli. Suppose for instance that you are both a mathematician and an industrialist, and want to transfer a huge amount of coal from your mines to your factories. You can hire trucks to do this transportation problem, but you have to pay them c(x, y) for each ton of coal which is transported from place x to place y. Both the amount of coal which you can extract from each mine, and the amount which each factory should receive, are fixed. As you are trying to solve the associated Monge-Kantorovich problem in order to minimize the price you have to pay, another mathematician comes to you and tells you “My friend, let me handle this for you: I will ship all your coal with my own trucks and you won’t have to worry about what goes where. I will just set a price ϕ(x) for loading one
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