1.2. Distance cost functions 37 Exercise 1.18 (Dual of C0,α). Let Ω be a smooth bounded open subset of Rn, let α (0, 1) and let E = C0,α(Ω) be the Banach space of α-H¨older continuous real-valued functions on Ω, equipped with the norm f E = sup |f(x)| + sup x=y |f(x) f(y)| |x y|α . Of course one can view the space M(Ω) as a subset of E∗, equipped with the norm E∗ = sup f E≤1 , f . Let μ and ν be two probability measures in E∗ show that (1.21) μ ν E∗ min ( μ ν TV , Tc(μ, ν) ) where c(x, y) = |x y|α. Remark 1.19. It is interesting to note that Tc above defines a norm which, roughly speaking, induces the topology of weak convergence of measures. We shall examine this in Chapter 7, and make precise statements there. 1.2.2. Transshipment. The Kantorovich-Rubinstein theorem implies that the total cost only depends on the difference μ ν. Thus, when the cost function is a metric, the Kantorovich optimal transportation problem is equivalent to the Kantorovich-Rubinstein transshipment problem: inf I[π] π[A × X] π[X × A] = ν)[A] . The condition appearing above should be compared to the condition for π Π(μ, ν), which is π[A × X] = μ[A], π[X × A] = ν[A]. For a general cost function, the transshipment problem is a strongly relaxed version of the transportation problem. For instance, in the case of a quadratic cost in Rn, the optimal transshipment cost between two given measures is in general 0. We shall not study the transshipment problem in this course, and refer to [211] for motivations and detailed study. Exercise 1.20. Give an interpretation of the Kantorovich-Rubinstein trans- shipment problem in (say) economics terms contrast this interpretation with that of the Monge-Kantorovich problem. Exercise 1.21 (Transshipment sometimes costs (almost) nothing). Let c(x, y) = |x y|2 in Rn, and let μ, ν be two probability measures on Rn, such that Tc(μ, ν) +∞. Let π Π(μ, ν) be any transference plan such that I[π] +∞. Of course this transference plan can also be considered as a transshipment plan, with an associated transshiping cost. In order to lower this transshipment cost, you wish to improve this plan, and you come up with the following strategy. Whenever x and y are some initial and final points, respectively, instead of shipping x to y, you ship x
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