Instability of the ray-monotone selector for $W_1$-optimal transport

Abstract

For the distance cost $c(x,y)=|x-y|$, the set $O(\mu ,\nu )$ of $W_1$-optimal plans is generally not a singleton. Under the classical absolute-continuity hypotheses in the Euclidean case, secondary variational selection by the quadratic energy $C_2$ yields the ray-monotone $W_1$-optimal plan. We provide a counterexample to an open problem posed by Santambrogio that concerns the stability of this selector under weak convergence of the marginals. More precisely, we construct a fixed absolutely continuous source $\mu$ and absolutely continuous targets ${\nu }_n\rightharpoonup \nu$ such that ${\gamma }^{\mathrm{sel}}(\mu ,{\nu }_n)\rightharpoonup {\gamma }^{\mathrm{hom}}$, where ${\gamma }^{\mathrm{hom}}\in O(\mu ,\nu )$ but ${\gamma }^{\mathrm{hom}}\ne {\gamma }^{\mathrm{sel}}(\mu ,\nu )$. We also identify the narrow Kuratowski limit of the optimal-plan sets $O(\mu ,{\nu }_n)$, derive the constrained $\Gamma$-limit for secondary energies of the form $\int \Phi (|x-y|)d\gamma$ with $\Phi \in C([0,2])$, and deduce a non-commutation result for the additive perturbation $c_{\epsilon }(x,y)=|x-y|+\epsilon |x-y{|}^2$.