  In this paper, we investigate properties of entropy-penalized Wasserstein
barycenters introduced by Bigot, Cazelles and Papadakis (2019) as a
regularization of Wasserstein barycenters first presented by Agueh and Carlier
(2011). After characterizing these barycenters in terms of a system of
Monge-Amp\`ere equations, we prove some global moment and Sobolev bounds as
well as higher regularity properties. We finally establish a central limit
theorem for entropic-Wasserstein barycenters.
