  We determine the optimal structure of couplings for the \emph{Martingale
transport problem} between radially symmetric initial and terminal laws $\mu,
\nu$ on $\R^d$ and show the uniqueness of optimizer. Here optimality means that
such solutions will minimize the functional $\E |X-Y|^p$ where $0<p \leq 1$,
and the dimension $d$ is arbitrary.
