It is true! Means that can be approximated by first taking harmonic and then arithmetic means are exactly the scalar Kubo-Ando means. A homogeneous mean m is of this type iff r \mapsto m(1,r)/r is operator monotone on. For m(x,y)=((x^p+y^p)/2)^{1/p}, this holds precisely when -1 \le p \le 1.

It is in the closure for p in {-1,0,1} so it must be true for all p in [-1,1]... or maybe not...

KL to a Gaussian target, (entropic) Wasserstein distance to a Gaussian. This invariance makes Gaussians an exceptionally handy test case.

Symmetric and positive (invertible) matrices.

Oui je pense

The course is currently running on Wednesdays, you should go if you are in Paris and interested in OT! www.college-de-france.fr/fr/personne/...

I am biased toward the SURE, I won’t take the risk to estimate without Stein.