  Consider the empirical measure, $\hat{\mathbb{P}}_N$, associated to $N$
i.i.d. samples of a given probability distribution $\mathbb{P}$ on the unit
interval. For fixed $\mathbb{P}$ the Wasserstein distance between
$\hat{\mathbb{P}}_N$ and $\mathbb{P}$ is a random variable on the sample space
$[0,1]^N$. Our main result is that its normalised quantiles are asymptotically
maximised when $\mathbb{P}$ is a convex combination between the uniform
distribution supported on the two points $\{0,1\}$ and the uniform distribution
on the unit interval $[0,1]$. This allows us to obtain explicit asymptotic
confidence regions for the underlying measure $\mathbb{P}$.
  We also suggest extensions to higher dimensions with numerical evidence.
