Characterization of measures on the real line that are critically unstable under small shifts

We study the perturbation of a measure $\mu \in \mathcal{P}(\mathbb{R})$ consisting in superposing two copies of $\mu$, each slightly shifted by a small distance $\pm h$. The difference between $\mu$ and its perturbation is measured with a Wasserstein distance. For any $\mu$, this distance is bounded from above by $h$. We show that measures for which this critical rate is achieved when $h$ goes to 0 are characterized as the ones giving most of their mass to some particular porous sets. This is used to identify which measures $\mu$ on the real line have a 2-Wasserstein tangent cone equal to the set of directions inducing curves with maximal initial speed.