Consider the optimal transport map $T$ between $N(\mu_0,\Sigma_0)$ and $N(\mu_1,\Sigma_1)$. I believed that the optimal transport was given by:
$$ T(x) = \mu_1 + \Sigma_1^{1/2} \Sigma_0^{-1/2}(x-\mu_0) $$
However in Peyre's book "Computational optimal transport" and other resources they claim that the map is:
$$T(x) = \mu_1 + (\Sigma_0^{-1/2}(\Sigma_0^{1/2}\Sigma_1 \Sigma_0^{1/2})^{1/2} \Sigma_0^{-1/2})(x-\mu_0) $$
I was wondering what is wrong with the transport I wrote.
My attempt
To prove that the proposed map works I used the Monge-Ampere equation. It is a straight forward calculation and everything works nicely.
Recall that for $T(x)$ to be optimal it has to be the gradient of a convex function $\phi(x).$ In your case if $T$ is the gradient of a function $\phi$ then $$ \Delta \phi = \Sigma_1^{1/2}\Sigma_0^{-1/2}, $$ but this matrix won't be in general positive definite or symmetric. Therefore you are considering cannot be optimal.