We study a class of functionals subject to a duality restriction. The functional is of the form $J(U, V)= \int_U f(x)dx +\int_V g(y)dy$, where $f, g$ are given non-negative functions. This model covers several geometric and physical applications, including the Minkowski problem in the sphere, and Kantorovich’s dual functional in optimal transport. The Euler equations of the functionals are of Monge-Amp\`{e}re type. In this talk, we discuss new methods and developments related to the functionals and their applications.