  We describe a construction of Stein kernels using moment maps, which are
solutions to a variant of the Monge-Amp\`ere equation. As a consequence, we
show how regularity bounds on these maps control the rate of convergence in the
classical central limit theorem, and derive new rates in
Kantorovitch-Wasserstein distance in the log-concave situation, with explicit
polynomial dependence on the dimension.
