Sharp bounds for max-sliced Wasserstein distances

We obtain essentially matching upper and lower bounds for the expected max-sliced 1-Wasserstein distance between a probability measure on a separable Hilbert space and its empirical distribution from $n$ samples. By proving a Banach space version of this result, we also obtain an upper bound, that is sharp up to a log factor, for the expected max-sliced 2-Wasserstein distance between a symmetric probability measure $μ$ on a Euclidean space and its symmetrized empirical distribution in terms of the operator norm of the covariance matrix of $μ$ and the diameter of the support of $μ$.