Structure of martingale transports in finite dimensions

We study the structure of martingale transports in finite dimensions. We consider the family $\mathcal{M}(\mu,\nu)$ of martingale measures on $\mathbb{R}^N \times \mathbb{R}^N$ with given marginals $\mu,\nu$, and construct a family of relatively open convex sets $\{C_x:x\in \mathbb{R}^N \}$, which forms a partition of $\mathbb{R}^N$, and such that any martingale transport in $\mathcal{M}(\mu,\nu)$ sends mass from $x$ to within $\overline{C_x}$, $\mu(dx)$--a.e. Our results extend the analogous one-dimensional results of M. Beiglb\"ock and N. Juillet (2016) and M. Beiglb\"ock, M. Nutz, and N. Touzi (2015). We conjecture that the decomposition is canonical and minimal in the sense that it allows to characterise the martingale polar sets, i.e. the sets which have zero mass under all measures in $\mathcal{M}(\mu,\nu)$, and offers the martingale analogue of the characterisation of transport polar sets proved in M. Beiglb\"ock, M. Goldstern, G. Maresch, and W. Schachermayer (2009).