  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.
