Suppose we have a probability distribution $\rho$ on $\mathbb{R}^d$. Let $ E \subset \operatorname{supp}(\rho) $, and $R_\theta$ a rotation of angle $\theta$ such that $ R_\theta E \subset \operatorname{supp}(\rho) $. Let $ \rho(x \mid x \in E) $ be the conditional distribution on $E$.

Is it true that $$ W_1(\rho(x \mid x \in E),\rho(x \mid x \in R_\theta E) \lesssim \operatorname{diam}(\operatorname{supp}(\rho)) \cdot \theta \qquad ? $$

I think this is true at least if $\rho$ is rotational invariant, but I was wondering if it can be generalized.

Thanks

Suppose we have a probability distribution $\rho$ on $\mathbb{R}^d$. Let $ E \subset \operatorname{supp}(\rho) $, and $R_\theta$ a rotation of angle $\theta$ such that $ R_\theta E \subset \operatorname{supp}(\rho) $. Let $ \rho(x \mid x \in E) $ be the conditional distribution on $E$.

Is it true that $$ W_1(\rho(x \mid x \in E),\rho(x \mid x \in R_\theta E)) \lesssim \operatorname{diam}(\operatorname{supp}(\rho)) \cdot \theta \qquad ? $$

I think this is true at least if $\rho$ is rotational invariant, but I was wondering if it can be generalized.

Thanks