Conditions for existence of single valued optimal transport maps on convex boundaries with nontwisted cost
Seonghyeon Jeong, Jun Kitagawa
TL;DR
This work proves the existence and regularity of Monge (single-valued) optimal transport maps on the boundary ∂Ω of a convex body Ω ⊂ ℝ^{n+1} for the ambient squared-distance cost c(X,X̄) = |X−X̄|^2/2, under μ, μ̄ supported on ∂Ω with densities bounded away from 0 and ∞ and with small MK_2 distance. The authors develop a localized QQConv condition, localized Aleksandrov estimates, and a stay-away mechanism, then approximate Ω by uniformly convex bodies to construct a dual potential with controlled Lipschitz constant; passing to the limit yields a Monge solution γ = (Id×T)_# μ with T continuous on ∂Ω, and Hölder regularity when ∂Ω is C^{1,α}. The results are sharp (nonexistence can occur for non-C^1 boundaries) and extend Monge map theory beyond spheres and strictly convex domains to general convex bodies with rough densities. The approach combines c-convexity, localization, and stability under Hausdorff convergence to handle non-strict convexity and discontinuous densities, providing quantitative regularity in terms of boundary smoothness.
Abstract
We prove that if $Ω\subset \mathbb{R}^{n+1}$ is a (not necessarily strictly) convex, $C^1$ domain, and $μ$ and $\barμ$ are probability measures absolutely continuous with respect to surface measure on $\partial Ω$, with densities bounded away from zero and infinity, whose $2$-Monge-Kantorovich distance is sufficiently small, then there exists a continuous Monge solution to the optimal transport problem with cost function given by the quadratic distance on the ambient space $\mathbb{R}^{n+1}$. This result is also shown to be sharp, via a counterexample when $Ω$ is uniformly convex but not $C^1$. Additionally, if $Ω$ is $C^{1, α}$ regular for some $α$, then the Monge solution is shown to be Hölder continuous.