Irreducible convex paving for decomposition of multidimensional martingale transport plans

Martingale transport plans on the line are known from Beiglböck and Juillet (Ann. Probab. 44 (2016) 42–106) 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 Rᵈ, d≥1. Our decomposition is a partition of Rᵈ 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.