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
> $$\{(x, \, y) \mid \ x, \, y \in \mathbb R^d, \ x/|x| = y/|y| \}?$$
> 
> 
> 



Is the optimal transport of radially symmetric measures also radially symmetric?
*This is my commnent (which nobody sees).*


The answer is "yes".
It follows since the quotient map $\mathbb{R}^d\to[0,\infty)$ is a submetry.
The optimal plan between pushforward measures on $[0,\infty)$ lifts uniquely to an optimal plan on $\mathbb{R}^d$


