  We provide a new proof of the known partial regularity result for the optimal
transportation map (Brenier map) between two sets. Contrary to the existing
regularity theory for the Monge-Amp{\`e}re equation, which is based on the
maximum principle, our approach is purely variational. By constructing a
competitor on the level of the Eulerian (Benamou-Brenier) formulation, we show
that locally, the velocity is close to the gradient of a harmonic function
provided the transportation cost is small. We then translate back to the
Lagrangian description and perform a Campanato iteration to obtain an
$\epsilon$-regularity result.
