Equivalence of ray monotonicity properties and classification of optimal transport maps for strictly convex norms

In this paper, we first define ray increasing and decreasing monotonicity of maps. If 𝑇 is an optimal transport map for the Monge problem with cost function \lVert y-x\rVert_{\mathrm{sc}} in R^{n} or 𝑇 is an optimal transport map for the Monge problem with cost function d(x,y), the geodesic distance, in more general, non-branching geodesic spaces 𝑋, we show respectively equivalence of some previously introduced monotonicity properties and the property of ray increasing as well as ray decreasing monotonicity which we define in this paper. Then, by solving secondary variational problems associated with strictly convex and concave functions respectively, we show that there exist ray increasing and decreasing optimal transport maps for the Monge problem with cost function \lVert y-x\rVert_{\mathrm{sc}}. Finally, we give the classification of optimal transport maps for the Monge problem such that the cost function \lVert y-x\rVert_{\mathrm{sc}} further satisfies the uniform smoothness and convexity estimates. That is, all of the optimal transport maps for such Monge problem can be divided into three different classes: the ray increasing map, the ray decreasing map and others.

More General Surplus

This chapter considers a case with a more general surplus function. It shows that when the scalar-product surplus is replaced by a more general function, much of the machinery put in place in Chapter 6 goes through. In particular, it is possible to generalize convex analysis in a natural way, and to obtain generalized notions of convex conjugates, of convexity, and of a subdifferential that are perfectly suited to the problem. A general result on the existence of dual minimizers is given, as well as sufficient conditions for the existence of a solution to the Monge problem.