Characterization of copulas with given diagonal and opposite diagonal sections

In recent years special attention has been devoted to the problem of finding a copula, the diagonal section and opposite diagonal section of which are known. For given diagonal function and opposite diagonal functions, we provide necessary and sufficient conditions for the existence of a copula to have these functions as diagonal and opposite diagonal sections. We make use of techniques related to interpolation between the diagonals, the construction of checkerboard copulas, and linear programming. This result allows us to solve two open problems: to characterize the class of copulas where the knowledge of diagonal and opposite diagonal sections determines the copula in a unique way, and to formulate necessary and sufficient conditions for each pair of such functions to be the diagonal and opposite diagonal sections of a unique copula.