  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.
