Let say I have $N$ discrete probability measures $(mu_1,…,mu_N)$ where each of them has $n$ points in $mathbb{R}^2$ of equal mass.

Let $P(mathcal{X})$ be the space of these probability measures on a Hilbert space $mathcal{X}$ (which is $mathbb{R}^2$ in our case if I’m not mistaken).

We can define the Wasserstein barycenter:

$overline{mu} = argmin_{nu in P(mathcal{X})} sum_i^N alpha_i W_2^2(mu_i, nu)$.

Under the 2-Wasserstein metric $W_2$ and with $alpha_i = frac{1}{N},_{i=1..N}$ (in our case).

I’m interested to grap the intuition with the tangent space at $overline{mu}$. Especially in the notion of orthogonality in this tangent space.

Let say I have a vector (or velocity) field $v_1 in L^2(overline{mu}, mathcal{X})$ (the tangent space at $overline{mu}$), containing $n$ vectors, that move all the $n$ masses of $overline{mu}$ in a direction $v_1^i,_{i=1..n}$ respectively.

What would it mean to have a velocity field $v_2 in L^2(overline{mu}, mathcal{X})$ orthogonal to $v_1$ in the $L^2(overline{mu}, mathcal{X})$ sense? It is just that we need to have $v_2^i perp v_1^i,_{i=1..n}$ in the $mathbb{R}^2$ sense?

I’m interested in any reference that could help (especially because I could need to cite something) and any explications.

Thank you very much.