Monge property and bounding multivariate probability distribution functions with given marginals and covariances

Multivariate probability distributions with given marginals are considered, along with linear functionals, to be minimized or maximized, acting on them. The functionals are supposed to satisfy the Monge or inverse Monge or some higher order convexity property, and they may be only partially known. Existing results in connection with Monge arrays are reformulated and extended in terms of linear programming dual feasible bases. Lower and upper bounds are given for the optimum value as well as for unknown coefficients of the objective function, based on the knowledge of some dual feasible bases and corresponding objective function coefficients. In the two- and three-dimensional cases dual feasible bases are obtained for the problem, where not only the univariate marginals but also the covariances of the pairs of random variables are known.