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 $\mu,\nu$ ordered with respect to a cone $\mathcal{F}$ of functions on $\Omega$ stable under maxima. Under the assumption that any $\mathcal{F}$-transport between $\mu$ and $\nu$ is local, we establish the existence of the finest partitioning of $\Omega$, depending only on $\mu,\nu$ and the cone $\mathcal{F}$, into $\mathcal{F}$-convex sets, called irreducible components, such that any $\mathcal{F}$-transport between $\mu$ and $\nu$ 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 $\mu$ and $\nu$ 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 Ob{\l}\'oj 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