Brenier’s theorem gives the unique squared-cost map between two densities
docmilanfar · x · 2026-07-22
This post states the core geometric form of Brenier’s theorem.
- Between two densities $p(x)$ and $q(y)$, Brenier’s theorem guarantees existence, uniqueness, and monotonicity of the optimal map under squared Euclidean cost.
- The map is written as $u=\nabla \phi$, sending $p$ to $q$ through a convex potential.
- The same thread frames it as a decomposition of a vector field into a convex-gradient component plus a measure-preserving rearrangement.
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