  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. In this paper, we
apply two classical covariance inequalities (the Brascamp-Lieb and Cram\'er-Rao
inequalities) to prove a sharp bound on the Lipschitz constant of the map that
arises from entropically regularized optimal transport. In the limit as the
regularization tends to zero, we obtain an elegant and short proof of
Caffarelli's original result. We also extend Caffarelli's theorem to the
setting in which the Hessians of the log-densities of the measures are bounded
by arbitrary positive definite commuting matrices.
