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2024

On Optimal Transport Maps Between 1 /d-Concave Densities

Carlier, Guillaume, Figalli, Alessio, Santambrogio, Filippo

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In this paper, we extend the scope of Caffarelli's contraction theorem, which provides a measure of the Lipschitz constant for optimal transport maps between log-concave probability densities in $\R^d$.

Our focus is on a broader category of densities, specifically those that are $\nicefrac{1}{d}$-concave and can be represented as $V^{-d}$, where $V$ is convex.
By setting appropriate conditions, we derive linear or sublinear limitations for the optimal transport map.
This leads us to a comprehensive Lipschitz estimate that aligns with the principles established in Caffarelli's theorem.

On Optimal Transport Maps Between 1 /d-Concave Densities · Around

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Polar factorization and monotone rearrangement of vector-valued functions
Yann Brenier · 1991
Earlier work this paper cites.
The regularity of mappings with a convex potential
Luis A. Caffarelli · 1992
Earlier work this paper cites.
Existence and uniqueness of monotone measure-preserving maps
Robert J. McCann · 1995
Earlier work this paper cites.
A remarkable measure preserving diffeomorphism between two convex bodies in 𝐑 n {\bf R}^{n}
S. Alesker, S. Dar, and V. Milman · 1999
Earlier work this paper cites.
Monotonicity properties of optimal transportation and the FKG and related inequalities
Luis A. Caffarelli · 2000
Earlier work this paper cites.

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Then

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