Optimal martingale transport between radially symmetric marginals in general dimensions

We determine the optimal structure of couplings for the Martingale transport problem between radially symmetric initial and terminal laws $\mu ,\nu$ on ${\mathbb{R}}^d$ and show the uniqueness of optimizer. Here optimality means that such solutions will minimize the functional $\mathbb{E}f(\left|\right|X-Y\left|\right|)$ where $f$ is concave and strictly increasing, and the dimension $d$ is arbitrary.