Optimal transport and the Monge problem

Remarks

Monge posed the problem in 1781, in his memoir on cuttings and fillings, for the cost $c(x, y) = \abs{x - y}$ in $\R^3$. His aim was to move a pile of soil into a hole of the same volume with least total work. The Monge problem may have no admissible map at all: if $\mu$ is a Dirac mass and $\nu$ is not, no $T$ satisfies $T_\#\mu = \nu$. Kantorovich introduced the relaxed problem in 1942; under the hypotheses above it always has a minimizer, by weak compactness of the set of plans and lower semicontinuity of the cost. For $X = Y = \R^n$, $c(x, y) = \abs{x - y}^2$, $\mu$ absolutely continuous and both measures with finite second moments, Brenier's theorem gives a unique optimal plan, induced by the map $T = \nabla \varphi$ for a convex function $\varphi$. Then $\varphi$ solves a Monge–Ampère equation in a weak sense, and the optimal cost defines the squared Wasserstein distance $W_2(\mu, \nu)^2$. For Monge's original cost $\abs{x - y}$ in $\R^n$ with $\mu$ absolutely continuous and both measures with finite first moments, optimal maps exist but are in general not unique. Evans and Gangbo proved existence under extra hypotheses in 1999, and later work removed them.