Brenier's theorem

Remarks

Yann Brenier proved the theorem in work on the polar factorization of vector fields, published in 1987 and 1991. The proof uses Kantorovich duality: since $\abs{x - y}^2 = \abs{x}^2 + \abs{y}^2 - 2 \ip{x}{y}$, an optimal plan maximizes $\int \ip{x}{y} \, d\pi$, and its support lies in the subdifferential of a convex function $\varphi$. A convex function is differentiable outside a Lebesgue null set. So absolute continuity of $\mu$ makes the subdifferential single-valued $\mu$-almost everywhere, and the plan is carried by the graph of $\nabla \varphi$. Example: if $\nu$ is the image of $\mu$ under $x \mapsto A x + b$ with $A$ symmetric positive definite, then $\varphi(x) = \frac12 \ip{Ax}{x} + \ip{b}{x}$ works, and the affine map is optimal. Absolute continuity cannot be dropped: if $\mu = \delta_0$ and $\nu$ is not a Dirac mass, no map pushes $\mu$ to $\nu$. When $\mu$ and $\nu$ have densities $f$ and $g$, the potential $\varphi$ solves the Monge–Ampère equation $g(\nabla \varphi) \det D^2 \varphi = f$ in a weak sense.

Going deeper

For a student

The theorem solves the Monge problem for the quadratic cost. It gives the polar factorization: a nondegenerate map $u$ pushing $\mu$ to $\nu$ factors as $u = \nabla \varphi \circ s$, where $\varphi$ is convex and $s$ preserves $\mu$. It is the basis of the Wasserstein geometry of probability measures, of displacement convexity, and of regularity theory for the Monge–Ampère equation through Caffarelli's work.