  We give a new proof of the Caffarelli contraction theorem, which states that
the Brenier optimal transport map sending the standard Gaussian measure onto a
uniformly log-concave probability measure is Lipschitz. The proof combines a
recent variational characterization of Lipschitz transport map by the second
author and Juillet with a convexity property of optimizers in the dual
formulation of the entropy-regularized optimal transport (or Schr{\"o}dinger)
problem.
