The Monge problem in $\mathbb R^d$: Variations on a theme

In a recent paper the authors proved that, under natural assumptions on the first marginal, the Monge problem in $\mathbb R^d$ for cost given by a general norm admits a solution. Although the basic idea of the proof is simple, it involves some complex technical results. Here we will give a proof of the result in the simpler case of uniformly convex norm and we will also use very recent results by other authors [1]. This allows 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 the particular case of this paper is original.