# dg.differential geometry – Orthogonality in Wasserstein tangent space for discrete measures with equal mass

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.