On optimal transport maps between $\frac{1}{d}$-concave densities

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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 \mathbb{R}^{d} . Our focus is on a broader category of densities, specifically those that are \frac{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.