In the last page of Villani's book "Optimal Transport, Old and New" there's written this: let $N=\Vert\cdot\Vert$ be a uniformly and smooth convex norm on $\mathbb{R}^n$ (i.e. $\exists\lambda,\Lambda>0$ such that $\lambda I_n\leq\nabla^2N^2\leq\Lambda I_n$). Then given two compactly supported and absolutely continuous measure (w.r.t. the Lebesgue one) there is a unique optimal transport map given by$$T(x)=x-\nabla(N^2)^*(-\nabla\psi(x)),\qquad\psi\mbox{ a $c$-concave function}.$$ Is there a source that states that the optimal transport map is this one in normed spaces? As far as I know usually for manifolds it takes the form of $\exp_x(-\nabla\psi(x))$ where $\exp$ in this case is replaced by $Id-\nabla(N^2)^*$ (it really seems the $\exp$ on the Euclidean space since it is a translation/shift). Also, what does that symbol mean? In order for the statement to make sense $\nabla(N^2)^*(-\nabla\psi(x))$ must be a vector on $\mathbb{R}^n$ and so $\nabla(N^2)^*$ a $(d\times d)$matrix: so does it symbolize the covariant differential?
Any source where the results are stated is very welcomed!