Introduction 1. Formulation of the optimal transportation problem Assume that we are given a pile of sand (say), and a hole that we have to completely fill up with the sand. Obviously, the pile and the hole must have the same volume. Let us normalize the mass of the pile to 1. We shall model both the pile and the hole by probability measures μ, ν, defined respectively on some measure spaces X and Y . Whenever A and B are measurable subsets of X and Y respectively, μ[A] gives a measure of how much sand is located inside A and ν[B] of how much sand can be piled in B. Moving the sand around needs some effort, which is modelled by a mea- surable cost function defined on X × Y . Informally, c(x, y) tells how much it costs to transport one unit of mass from location x to location y. It is natural to assume at least that c is measurable and nonnegative. One should not a priori exclude the possibility that c takes infinite values, and so c should be a measurable map from X × Y to R ∪ {+∞}. X Y μ ν x y Figure 1. The mass transportation problem 1

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