Let $\mu$ and $\nu$ be radially symmetric probability measures on $\mathbb R^d$. Consider the Kantorovich optimal transport problem between $\mu$ and $\nu$, with convex, nonnegative cost. Suppose there exists at least an optimal transport plan between $\mu$ and $\nu$ with finite cost.

Question.Does it follow that any optimal transport plan $\gamma \in \mathcal P (\mathbb R^d \times \mathbb R^d)$ is radially symmetric, in the sense that $\gamma$ is concentrated on the set $$\{(x, \, y) \mid \ x, \, y \in \mathbb R^d, \ x/|x| = y/|y| \}?$$

3more comments