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arXiv · 2404.05456

On Optimal Transport Maps Between 1 /d-Concave Densities

Guillaume Carlier·Alessio Figalli·Filippo Santambrogio
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Abstract

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.

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Guillaume Carlier, Alessio Figalli, Filippo Santambrogio. 2024-04-08. On Optimal Transport Maps Between 1 /d-Concave Densities. https://arxiv.org/abs/2404.05456

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