Stability of Brenier maps

For the uniform measure on a convex body, quadratic optimal transport maps depend on the target in a one-third Hölder way for the Wasserstein distance, with one constant for all targets in a fixed compact set. The exponent one third is optimal, already on a cube with three-atom targets, which disproves a conjectured square-root estimate. The proof: derivatives of Laguerre cells; a uniform estimate of moment changes by mass changes, from a midpoint coupling by Knothe's rearrangement; moving the sites of finite targets with fixed masses keeps the potentials within O(W_2); a slice interpolation inequality turns this into the exponent one third. Proved in full in the programme, with no named hypothesis. The course is an exposition of OpenAI's proof (2026).

Five lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Optimal maps of finite affine maxima
2. Laguerre cells and prescribed masses
3. A uniform estimate for cell moments
4. Moving sites with fixed masses
5. From potentials to maps

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Measure theory: Fubini and Tonelli, dominated convergence, image measures (core course Measure and Integration); the implicit function theorem and the Arzelà–Ascoli theorem (core course Real Analysis II).