# Irreducible convex paving for decomposition of multi-dimensional martingale transport plans

## TL;DR

This paper extends the irreducible decomposition of martingale transport plans from one dimension to higher dimensions, providing a convex partition of R^d and analyzing the structure of polar sets.

## Contribution

It introduces a convex decomposition for martingale transport plans in R^d, generalizing the one-dimensional case and establishing the existence of plans filling these components.

## Key findings

- Decomposition into convex components in R^d
- Existence of martingale plans for each component
- Characterization of polar sets with respect to martingale plans

## Abstract

Martingale transport plans on the line are known from Beiglbock & Juillet to have an irreducible decomposition on a (at most) countable union of intervals. We provide an extension of this decomposition for martingale transport plans in R^d, d larger than one. Our decomposition is a partition of R^d consisting of a possibly uncountable family of relatively open convex components, with the required measurability so that the disintegration is well-defined. We justify the relevance of our decomposition by proving the existence of a martingale transport plan filling these components. We also deduce from this decomposition a characterization of the structure of polar sets with respect to all martingale transport plans.