Strong equivalence between metrics of Wasserstein type

The sliced Wasserstein metric W-p. and more recently max-sliced Wasserstein metric (W) over bar (p) have attracted abundant attention in data sciences and machine learning due to their advantages to tackle the curse of dimensionality, see e.g. [15], [6]. A question of particular importance is the strong equivalence between these projected Wasserstein metrics and the (classical) Wasserstein metric W-p. Recently, Paty and Cuturi have proved in [14] the strong equivalence of (W) over bar (2) and W-2. We show that the strong equivalence also holds for p = 1, while the sliced Wasserstein metric does not share this nice property.