In some recent papers it is proved that, under natural assumptions on the first marginal, the Monge problem in the metric space Rd equipped with a general norm admits a solution. Although the basic idea of the solution is simple the proof involves some very complex technical results. Here we will report a proof of the result in the simpler case of uniformly convex norms. Uniform convexity allow us to reduce the technical burdens while still giving the main ideas of the general proof. The proof of the density of the transport set given in this paper is original.