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Sharp Stability of Brenier Potentials and its Link to Poincaré Inequalities

  • Cyril Letrouit
  • Tudor Manole
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Abstract

We prove a new functional inequality for uniformly log-concave measures, stronger than the Poincaré inequality. This new inequality bounds the variance of differences of convex functions and is motivated by the stability theory of optimal transport: for a fixed uniformly log-concave source measure, it provides a sharp stability bound for Brenier potentials with respect to variations of the target measure. Applying this inequality to Gaussian measures, we recover the fact that the Bures-Wasserstein distance between two positive semidefinite matrices is equivalent to the Frobenius distance between their square roots. Our proofs hinge upon linearization arguments for the Monge-Ampère equation governing the quadratic optimal transport problem.

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Cite

BibTeXRIS
BibTeX
@misc{letrouit2026sharp,
  title = {{Sharp Stability of Brenier Potentials and its Link to Poincaré Inequalities}},
  author = {Cyril Letrouit and Tudor Manole},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37659},
  eprint = {2610.37659},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37659},
}
RIS
TY  - UNPB
TI  - Sharp Stability of Brenier Potentials and its Link to Poincaré Inequalities
AU  - Cyril Letrouit
AU  - Tudor Manole
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37659
UR  - https://arxiv.org/abs/2610.37659
AB  - We prove a new functional inequality for uniformly log-concave measures, stronger than the Poincaré inequality. This new inequality bounds the variance of differences of convex functions and is motivated by the stability theory of optimal transport: for a fixed uniformly log-concave source measure, it provides a sharp stability bound for Brenier potentials with respect to variations of the target measure. Applying this inequality to Gaussian measures, we recover the fact that the Bures-Wasserstein distance between two positive semidefinite matrices is equivalent to the Frobenius distance between their square roots. Our proofs hinge upon linearization arguments for the Monge-Ampère equation governing the quadratic optimal transport problem.
ER  -