  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).
