Localisation for constrained transports I: theory

Abstract

We investigate an analogue of the irreducible convex paving in the context of generalised convexity. Consider two Radon probability measures $μ,ν$ ordered with respect to a cone $\mathcal{F}$ of functions on $Ω$ stable under maxima. Under the assumption that any $\mathcal{F}$-transport between $μ$ and $ν$ is local, we establish the existence of the finest partitioning of $Ω$, depending only on $μ,ν$ and the cone $\mathcal{F}$, into $\mathcal{F}$-convex sets, called irreducible components, such that any $\mathcal{F}$-transport between $μ$ and $ν$ must adhere to this partitioning. Furthermore, we demonstrate that a set, whose sections are contained in the corresponding irreducible components, is a polar set with respect to all $\mathcal{F}$-transports between $μ$ and $ν$ if and only if it is a polar set with respect to all transports. This provides an affirmative answer to a generalisation of a conjecture proposed by Obłój and Siorpaes regarding polar sets in the martingale transport setting. Among our contributions is also a generalisation of the Strassen's theorem to the setting of generalised convexity