A proof of Sudakov theorem with strictly convex norms

Abstract

he paper establishes a solution to the Monge problem in ℝ for a possibly asymmetric norm cost function and absolutely continuous initial measures, under the assumption that the unit ball is strictly convex-but not necessarily differentiable nor uniformly convex. The proof follows the strategy initially proposed by Sudakov in 1976, found to be incomplete in 2000; the missing step is fixed in the above case adapting a disintegration technique introduced for a variational problem. By strict convexity, mass moves along rays, and we also investigate the divergence of the vector field of rays

A User’s Guide to Optimal Transport

“…25). Then there exist unique s ∈ S(Ω) and ∇ϕ, with ϕ convex, such that S = (∇ϕ) • s. Also, s is the unique minimizer of |S −s| 2 dµ, among alls ∈ S(Ω).…”

“…Later, existence of optimal maps for the case c(x, y) := x − y , · being any norm has been established, at increasing levels of generality, in [9], [28], [27] (containing the most general result, for any norm) and [25].…”

The disintegration of the Lebesgue measure on the faces of a convex function

“…This kind of strategy was first used in order to disintegrate the Lebesgue measure on a collection of disjoint segments in [7], and then in [8].…”

“…This fundamental estimate, as in [7,8], is proved approximating the 1-dimensional faces with a sequence of finitely many cones with vertex in C ∩ H h − and basis in C ∩ H t .…”

“…The basis of the technique we use was first presented in order to solve a variational problem in [7] and it has been successfully applied to the existence of optimal transport maps for strictly convex norms in [8].…”