Solution of the Monge-Kantorovich problem for a class of functionals. (Q1432187)

The functional \(\rho(t, P, Q)=\inf_{\xi, \eta}P(r(\xi, \eta)>t)\) is considered, where the infimum is taken over all random variables \(\xi\) and \(\eta\) given on the same probability space and having distributions \(P\) and \(Q\), respectively, and where \(r\) is certain kernel function. The main result determines the value of \(\rho(t, P, Q).\) The construction of a corresponding closest random variable is also included.