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The optimal transport map between the standard Gaussian measure and an $\alpha$-strongly log-concave probability measure is $\alpha^{-1/2}$-Lipschitz, as first observed in a celebrated theorem of Caffarelli.
Convex analysis
R. Tyrrell Rockafellar · 1970
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I. Csiszár · 1975
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On a formula for the L 2 L^{2} Wasserstein metric between measures on Euclidean and Hilbert spaces
Matthias Gelbrich · 1990
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Monotonicity properties of optimal transportation and the FKG and related inequalities
Luis A. Caffarelli · 2000
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