On the optimal mapping of distributions

We consider the problem of mappingX→Y, whereX andY have given distributions, so as to minimize the expected value of ∣X–Y∣2. This is equivalent to finding the joint distribution of the random variable (X, Y), with specified marginal distributions forX andY, such that the expected value of ∣X–Y∣2 is minimized. We give a sufficient condition for the minimizing joint distribution and supply numerical results for two special cases.