Best approximation problems relating to Monge–Kantorovich duality

Problems of the best approximation of bounded continuous functions on a topological space $X\times X$ by functions of the form $u(x)-u(y)$ are considered. Formulae for the values of the best approximations are obtained and the equivalence between the existence of precise solutions and the non-emptiness of the constraint set of the auxiliary dual Monge–Kantorovich problem with a special cost function is established. The form of precise solutions is described in terms relating to the Monge–Kantorovich duality, and for several classes of approximated functions the existence of precise solutions with additional properties, such as smoothness and periodicity, is proved.
Bibliography: 20 titles.