Optimal martingale transport between radially symmetric marginals in general dimensions

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 $\text{E}\mathrm{\mid }X-Y{\mathrm{\mid }}^p$ where $0<p\le 1$, and the dimension $d$ is arbitrary.