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

Abstract

Abstract 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 ∥ y - x ∥ sc \lVert y-x\rVert_{\mathrm{sc}} in R n R^{n} or 𝑇 is an optimal transport map for the Monge problem with cost function d ⁢ ( x , y ) 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 ∥ y - x ∥ sc \lVert y-x\rVert_{\mathrm{sc}} . Finally, we give the classification of optimal transport maps for the Monge problem such that the cost function ∥ y - x ∥ sc \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.