I just want to verify that my reasoning here is correct. It feels very basic but I can't seem to find this result in any textbook. It is well-known that the 1-Wasserstein distance in 1D can be expressed in closed form using cdfs: $W_1(\mu, \nu) = \int_\mathbb{R} {|F(x) - G(x)| dx}$. I am now interested in the Barycenter, i.e. the solution to
$\min \sum_{i=1}^nW_1(\mu, \mu_i)$
for given $\mu_i$. In my opinion, we can just rewrite the objective to $\int_\mathbb{R}\sum_{i=1}^n |F_\mu(x)-F_{\mu_i}(x)|dx$ and then see that the integrand is minimized pointwise if $F_\mu(x) = median(F_{\mu_1}(x), ..., F{\mu_n}(x))$. Now this should again give us a valid cdf and hence solve the problem.
Now, while the $W_2$-Barycenter appears in a number of textbooks, I didn't find this one. It would be good if anyone could give me either a reference so that I can cite it or tell me where I went wrong or confirm that it all seems good (maybe people just don't care enough about this case to include it in their books?).
