2. Basic questions 7 •But if p = 1, even the assumption that μ and ν do not give positive mass to small sets will not ensure that the solutions to the Monge and Kantorovich problems coincide (one can construct counter-examples concentrated on a set of dimension n ε, ε as small as desired). If μ and ν are absolutely continuous with respect to the Lebesgue measure, then there are solutions of the Monge problem which are also solutions of the Kantorovich problem however it is easy to check that there is no uniqueness. For p 1, the situation is again different: there is in general no solution of the Monge problem, except if μ and ν are concentrated on disjoint sets. And when they exist, these solutions are locally orientation-reversing, which is the opposite of what happens for p 1. These various examples were mainly extracted from [14, 141, 191]. Another illustration of the different geometrical properties of the cases p = 1 and p = 2 is provided by Problem 1 in Chapter 10. As for Question 2, it is less sensitive to the cost. But it does depend on it, as shown by the following examples. Consider, in a complete separable metric space (X, d), the cost c(x, y) = d(x, y)p, for some given p (0, ∞). Then (12) Tcmin(1,1/p) = inf π∈Π(μ,ν) X×X d(x, y)p dπ(x, y) min(1,1/p) is a metric on the space P (X), which metrizes weak convergence of probabil- ity measures, as soon as one controls the moments of order p. Here is a more precise statement: let (μk)k∈Æ be a sequence of probability measures such that for some x0 X, the sequence of measures dμk(x) d(x, x0)p is tight then μk converges to μ weakly if and only if Tc(μk,μ) −→ 0. We recall that a family ρk of nonnegative measures on a topological space X is said to be tight if for all ε 0 there is a compact set with supk ρk[X \ Kε] ε. On the other hand, consider the special cost c(x, y) = 1x=y (which is a peculiar distance!). Then the total transportation cost is a familiar object: (13) Tc(μ, ν) = 1 2 μ ν TV , with the subscript TV standing for “total variation”. This identity is a particular case of Strassen’s theorem it is often stated in the probabilistic version (14) inf E [X = Y ] = sup μ[F ] ν[F ] F closed . Of course, it implies that Tc metrizes the strong topology on the space of measures.
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