Locality of centred tangent cones in the Wasserstein space

The geometric tangent cone to a probability measure $\mu$ is a set of measure-valued applications that are almost geodesics. This is a nonlocal condition, typically lost when conditioning the measure on a given set. We show that if one removes the barycenter of any element of the tangent cone, then the resulting set of centred measure fields is characterized by a local condition. Precisely, centred tangent fields must be concentrated on a family of vector subspaces attached to any point, and these subspaces correspond to the normal spaces to some sets of ``dimension $k$'' on which the measure $\mu$ is concentrated.