Let $A$ be a positive-definite symmetric matrix. Consider the ellipsoid $E = \{ x \in \mathbb{R}^n \colon <x A^{-1} x> \leq 1 \}$.
Now consider uniform distribution $\mu_1, \ldots, \mu_n$ on $E_n = \{x \in \mathbb{R}^n \colon <x A_n^{-1} x> \leq 1 \}$. (for $1 \leq i \leq n$, $\mu_i$ uniform on $E_i$.)
What's the 2-Wasserstein barycenter of $\mu_1, \ldots, \mu_n$?
(where $W_2(\mu_1, \mu_2)^2 = \inf_{\pi \in \Gamma(\mu_1, \mu_2)} \int d(x,y)^2d \pi(x,y)$, and the barycenter is the measure that minimizes the sum $\sum_{i=1}^n W_2(\nu, \mu_i)^2$ )
Thanks!
It seems like it should be some uniform distribution on an ellipsoid, but I can't seem to figure it out.