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Transportation onto log-Lipschitz perturbations
[Submitted on 5 May 2023 (v1), last revised 24 Aug 2026 (this ve · 2023-05-06 · via math updates on arXiv.org

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Abstract:We establish sufficient conditions for the existence of globally Lipschitz transport maps between probability measures and their log-Lipschitz perturbations, with dimension-free bounds. Our results include Gaussian measures on Euclidean spaces and uniform measures on spheres as source measures. More generally, we prove results for source measures on manifolds satisfying strong curvature assumptions. These seem to be the first examples of dimension-free Lipschitz transport maps in non-Euclidean settings, which are moreover sharp on the sphere. We also present some applications to functional inequalities, including a new dimension-free Gaussian isoperimetric inequality for log-Lipschitz perturbations of the standard Gaussian measure. Our proofs are based on the Langevin flow construction of transport maps of Kim and Milman.

Submission history

From: Yair Shenfeld [view email]
[v1] Fri, 5 May 2023 18:37:24 UTC (23 KB)
[v2] Mon, 11 Dec 2023 15:27:18 UTC (24 KB)
[v3] Mon, 24 Aug 2026 20:17:40 UTC (24 KB)